MATHEMATICAL COMBINATORICS (INTERNATIONAL BOOK SERIES), Volume 1 / 2013

8/9/2019 MATHEMATICAL COMBINATORICS (INTERNATIONAL BOOK SERIES), Volume 1 / 2013

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Vol.1, 2013 ISBN 978-1-59973-218-3

Mathematical Combinatorics(International Book Series)

Edited By Linfan MAO

The Madis of Chinese Academy of Sciences and

Beijing University of Civil Engineering and Architecture

March, 2013





Editorial Board (2nd)

Editor-in-Chief

Linfan MAOChinese Academy of Mathematics and SystemScience, P.R.ChinaandBeijing University of Civil Engineering andArchitecture, P.R.ChinaEmail: [email protected]

Deputy Editor-in-Chief

Guohua SongBeijing University of Civil Engineering andArchitecture, P.R.ChinaEmail: [email protected]

Editors

S.BhattacharyaDeakin UniversityGeelong Campus at Waurn Ponds

AustraliaEmail: [email protected]

Dinu BratosinInstitute of Solid Mechanics of Romanian Ac-ademy, Bucharest, Romania

Junliang CaiBeijing Normal University, P.R.ChinaEmail: [email protected]

Yanxun ChangBeijing Jiaotong University, P.R.ChinaEmail: [email protected]

Jingan CuiBeijing University of Civil Engineering andArchitecture, P.R.ChinaEmail: [email protected]

Shaofei DuCapital Normal University, P.R.ChinaEmail: [email protected]

Baizhou He

Beijing University of Civil Engineering andArchitecture, P.R.ChinaEmail: [email protected]

Xiaodong HuChinese Academy of Mathematics and SystemScience, P.R.ChinaEmail: [email protected]

Yuanqiu HuangHunan Normal University, P.R.ChinaEmail: [email protected]

H.IseriManseld University, USAEmail: [email protected]

Xueliang LiNankai University, P.R.ChinaEmail: [email protected]

Guodong LiuHuizhou UniversityEmail: [email protected]

Ion PatrascuFratii Buzesti National CollegeCraiova Romania

Han RenEast China Normal University, P.R.ChinaEmail: [email protected]

Ovidiu-Ilie SandruPolitechnica University of BucharestRomania.

Tudor SireteanuInstitute of Solid Mechanics of Romanian Ac-ademy, Bucharest, Romania.

W.B.Vasantha KandasamyIndian Institute of Technology, IndiaEmail: [email protected]



ii International Journal of Mathematical Combinatorics

Luige VladareanuInstitute of Solid Mechanics of Romanian Ac-ademy, Bucharest, Romania

Mingyao XuPeking University, P.R.ChinaEmail: [email protected]

Guiying YanChinese Academy of Mathematics and SystemScience, P.R.ChinaEmail: [email protected]

Y. ZhangDepartment of Computer ScienceGeorgia State University, Atlanta, USA

Famous Words:

Mathematics, rightly viewed, posses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture.

By Bertrand Russell, an England philosopher and mathematician.



Math.Combin.Book Ser. Vol.1(2013), 01-37

Global Stability of Non-Solvable

Ordinary Differential Equations With Applications

Linfan MAO

Chinese Academy of Mathematics and System Science, Beijing 100190, P.R.China

Beijing University of Civil Engineering and Architecture, Beijing 100045, P.R.China

E-mail: [email protected]

Abstract : Different from the system in classical mathematics, a Smarandache system isa contradictory system in which an axiom behaves in at least two different ways within thesame system, i.e., validated and invalided, or only invalided but in multiple distinct ways.

Such systems exist extensively in the world, particularly, in our daily life. In this paper, wediscuss such a kind of Smarandache system, i.e., non-solvable ordinary differential equationsystems by a combinatorial approach, classify these systems and characterize their behaviors,particularly, the global stability, such as those of sum-stability and prod-stability of suchlinear and non-linear differential equations. Some applications of such systems to othersciences, such as those of globally controlling of infectious diseases, establishing dynamicalequations of instable structure, particularly, the n-body problem and understanding globalstability of matters with multilateral properties can be also found.

Key Words : Global stability, non-solvable ordinary differential equation, general solution,G-solution, sum-stability, prod-stability, asymptotic behavior, Smarandache system, inherit

graph, instable structure, dynamical equation, multilateral matter.AMS(2010) : 05C15, 34A30, 34A34, 37C75, 70F10, 92B05

§1. Introduction

Finding the exact solution of an equation system is a main but a difficult objective unless somespecial cases in classical mathematics. Contrary to this fact, what is about the non-solvable case for an equation system? In fact, such an equation system is nothing but a contradictorysystem, and characterized only by having no solution as a conclusion. But our world is overlapand hybrid. The number of non-solvable equations is much more than that of the solvable

and such equation systems can be also applied for characterizing the behavior of things, whichreect the real appearances of things by that their complexity in our world. It should be notedthat such non-solvable linear algebraic equation systems have been characterized recently bythe author in the reference [7]. The main purpose of this paper is to characterize the behaviorof such non-solvable ordinary differential equation systems.

1 Received November 16, 2012. Accepted March 1, 2013.



2 Linfan Mao

Assume m, n ≥ 1 to be integers in this paper. Let

X = F (X ) (DES 1)

be an autonomous differential equation with F : R n

→ R n and F (0) = 0, particularly, let

X = AX (LDES 1)

be a linear differential equation system and

x(n ) + a1x(n − 1) + · · ·+ an x = 0 (LDE n )

a linear differential equation of order n with

A =

a11 a12 · · · a1n

a21 a22

· · · a2n

· · · · ·· · · · · ··an 1 an 2 · · · ann

X =

x1(t)

x2(t)

· · ·xn (t)

and F (t, X ) =

f 1(t, X )

f 2(t, X )

· · ·f n (t, X )

,

where all a i , aij , 1 ≤ i, j ≤ n are real numbers with

X = ( x1 , x2 , · · · , xn )T

and f i (t) is a continuous function on an interval [ a, b] for integers 0 ≤ i ≤ n. The followingresult is well-known for the solutions of ( LDES 1) and ( LDE n ) in references.

Theorem 1.1([13]) If F (X ) is continuous in

U (X 0) : |t −t0| ≤ a, X −X 0 ≤ b (a > 0, b > 0)

then there exists a solution X (t) of differential equation (DES 1) in the interval |t −t0| ≤ h ,where h = min {a, b/M }, M = max

( t,X ) U ( t 0 ,X 0 )F (t, X ) .

Theorem 1.2([13]) Let λi be the ki -fold zero of the characteristic equation

det( A −λI n × n ) = |A −λI n × n | = 0

or the characteristic equation

λn + a1λn − 1 + · · ·+ an − 1λ + an = 0

with k1 + k2 + · · ·+ ks = n. Then the general solution of (LDES 1) is

n

i=1

ci β i (t)eα i t ,



Global Stability of Non-Solvable Ordinary Differential Equations With Applications 3

where, ci is a constant, β i (t) is an n-dimensional vector consisting of polynomials in t deter-mined as follows

β 1(t) =

t11

t21

· · ·tn 1

β 2(t) =

t11 t + t12

t21 t + t22

· · · · · · · · ·tn 1 t + tn 2

· · · · · · · · · · · · · · · · · · · · · · · · · · ·

β k1 (t) =

t 11(k1 − 1)! tk1 − 1 + t12

(k1 − 2)! tk1 − 2 +

· · ·+ t1k1

t 21(k1 − 1)! tk1 − 1 + t22

(k1 − 2)! tk1 − 2 + · · ·+ t2k1

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·t n 1

(k1 − 1)! tk1 − 1 + tn 2(k1 − 2)! tk1 − 2 + · · ·+ tnk 1

β k1 +1 (t) =

t1( k1 +1)

t2( k1 +1)

· · · · · ·tn (k1 +1)

β k1 +2 (t) =

t11 t + t12

t21 t + t22

· · · · · · · · ·tn 1 t + tn 2

· · · · · · · · · · · · · · · · · · · · · · · · · · ·

β n (t) =

t 1( n − k s +1)(k s − 1)! tk s − 1 + t1( n − k s +2)

(k s − 2)! tk s − 2 + · · ·+ t1nt 2( n − k s +1)

(k s − 1)! tk s − 1 + t2( n − k s +2)(k s − 2)! tk s − 2 + · · ·+ t2n

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·t n ( n − k s +1)

(k s − 1)! tk s − 1 + tn ( n − k s +2)(k s − 2)! tk s − 2 + · · ·+ tnn

with each tij a real number for 1

≤ i, j

≤ n such that det([ t ij ]n × n )

= 0 ,

α i =

λ1 , if 1 ≤ i ≤ k1 ;

λ2 , if k 1 + 1 ≤ i ≤ k2 ;

· · · · · · · ·· · · ·· · · · ·· · · · ;

λ s , if k 1 + k2 + · · ·+ ks − 1 + 1 ≤ i ≤ n.



4 Linfan Mao

The general solution of linear differential equation (LDE n ) is s

i =1

(ci 1 tk i − 1 + ci 2tk i − 2 + · · ·+ ci (k i − 1) t + cik i )eλ i t ,

with constants cij , 1 ≤ i ≤ s, 1 ≤ j ≤ ki .Such a vector family β i (t)eα i t , 1 ≤ i ≤ n of the differential equation system ( LDES 1) and

a family t leλ i t , 1 ≤ l ≤ ki , 1 ≤ i ≤ s of the linear differential equation ( LDE n ) are called thesolution basis , denoted by

B = { β i (t)eα i t | 1 ≤ i ≤ n } or C = { t l eλ i t | 1 ≤ i ≤ s, 1 ≤ l ≤ ki }.

We only consider autonomous differential systems in this paper. Theorem 1 .2 implies thatany linear differential equation system ( LDES 1) of rst order and any differential equation(LDE n ) of order n with real coefficients are solvable. Thus a linear differential equation systemof rst order is non-solvable only if the number of equations is more than that of variables, and

a differential equation system of order n ≥ 2 is non-solvable only if the number of equationsis more than 2. Generally, such a contradictory system, i.e., a Smarandache system [4]-[6] isdened following.

Denition 1.3([4]-[6]) A rule R in a mathematical system (Σ; R) is said to be Smarandachely denied if it behaves in at least two different ways within the same set Σ, i.e., validated and invalided, or only invalided but in multiple distinct ways.

A Smarandache system (Σ; R) is a mathematical system which has at least one Smaran-dachely denied rule R.

Generally, let (Σ 1;R1 ) (Σ 2 ;R2), · · · , (Σ m ;Rm ) be mathematical systems, where Ri is arule on Σ i for integers 1

≤ i

≤ m. If for two integers i,j, 1

≤ i, j

≤ m, Σ i

= Σ j or Σ i = Σ j but

Ri = Rj , then they are said to be different , otherwise, identical . We also know the conceptionof Smarandache multi-space dened following.

Denition 1.4([4]-[6]) Let (Σ 1 ;R1), (Σ 2 ;R2), · · ·, (Σ m ;Rm ) be m ≥ 2 mathematical spaces,

different two by two. A Smarandache multi-space Σ is a union m

i =1Σ i with rules R =

m

i=1 Ri on

Σ , i.e., the rule Ri on Σ i for integers 1 ≤ i ≤ m, denoted by Σ; R .

A Smarandache multi-space Σ; R inherits a combinatorial structure, i.e., a vertex-edgelabeled graph dened following.

Denition 1.5([4]-[6]) Let Σ; R be a Smarandache multi-space with Σ =m

i =1Σ i and R =

m

i=1 Ri . Its underlying graph G Σ , R is a labeled simple graph dened by

V G Σ , R = {Σ 1 , Σ 2 , · · · , Σ m },

E G Σ , R = { (Σ i , Σ j ) | Σ i Σ j = , 1 ≤ i, j ≤ m}




with an edge labeling

lE : (Σ i , Σ j ) E G S, R → lE (Σ i , Σ j ) = Σ i Σ j ,

where is a characteristic on Σ i Σ j such that Σ i Σ j is isomorphic to Σ k Σ l if and only if (Σ i Σ j ) = (Σ k Σ l ) for integers 1 ≤ i, j, k,l ≤ m.

Now for integers m, n ≥ 1, let

X = F 1 (X ), X = F 2(X ), · · · , X = F m (X ) (DES 1m )

be a differential equation system with continuous F i : R n → R n such that F i (0) = 0, particu-larly, let

X = A1X, · · · , X = Ak X, · · · , X = Am X (LDES 1m )

be a linear ordinary differential equation system of rst order and

x(n )

+ a[0]

11 x(n − 1)

+ · · ·+ a[0]

1n x = 0x(n ) + a [0]

21 x(n − 1) + · · ·+ a [0]2n x = 0

· · · · · · · · · · · ·x(n ) + a [0]

m 1x(n − 1) + · · ·+ a [0]mn x = 0

(LDE nm )

a linear differential equation system of order n with

Ak =

a [k ]11 a[k ]

12 · · · a[k ]1n

a [k ]21 a[k ]

22 · · · a[k ]2n

· · · · ·· · · · · ··a [k ]

n 1 a[k ]n 2

· · · a[k]

nn

and X =

x1(t)

x2(t)

· · ·xn (t)

where each a [k ]ij is a real number for integers 0 ≤ k ≤ m, 1 ≤ i, j ≤ n.

Denition 1.6 An ordinary differential equation system (DES 1m ) or (LDES 1

m ) (or (LDE nm ))

are called non-solvable if there are no function X (t) (or x(t)) hold with (DES 1m ) or (LDES 1

m )(or (LDE n

m )) unless the constants.

The main purpose of this paper is to nd contradictory ordinary differential equationsystems, characterize the non-solvable spaces of such differential equation systems. For suchobjective, we are needed to extend the conception of solution of linear differential equations inclassical mathematics following.

Denition 1.7 Let S 0i be the solution basis of the ith equation in (DES 1m ). The -solvable, -

solvable and non-solvable spaces of differential equation system (DES 1m ) are respectively dened

by m

i=1

S 0i ,m

i=1

S 0i andm

i=1

S 0i −m

i =1

S 0i ,

where S 0i is the solution space of the ith equation in (DES 1m ).



6 Linfan Mao

According to Theorem 1 .2, the general solution of the ith differential equation in ( LDES 1m )

or the ith differential equation system in ( LDE nm ) is a linear space spanned by the elements

in the solution basis B i or C i for integers 1 ≤ i ≤ m. Thus we can simplify the vertex-edgelabeled graph G ,

R replaced each

i by the solution basis B i (or C i ) and

i j by

B i B j (or C i C j ) if B i B j = (or C i C j = ) for integers 1 ≤ i, j ≤ m. Such a vertex-edge labeled graph is called the basis graph of (LDES 1

m ) ((LDE nm )), denoted respectively by

G[LDES 1m ] or G[LDE n

m ] and the underlying graph of G[LDES 1m ] or G[LDE n

m ], i.e., clearedaway all labels on G[LDES 1

m ] or G[LDE nm ] are denoted by G[LDES 1

m ] or G[LDE nm ].

Notice thatm

i=1S 0i =

m

i=1S 0i , i.e., the non-solvable space is empty only if m = 1 in

(LDEq ). Thus G[LDES 1] K 1 or G[LDE n ] K 1 only if m = 1. But in general, thebasis graph G[LDES 1

m ] or G[LDE nm ] is not trivial. For example, let m = 4 and B 0

1 =

{eλ 1 t , eλ 2 t , eλ 3 t}, B 02 = {eλ 3 t , eλ 4 t , eλ 5 t }, B 0

3 = {eλ 1 t , eλ 3 t , eλ 5 t } and B 04 = {eλ 4 t , eλ 5 t , eλ 6 t },

where λi , 1 ≤ i ≤ 6 are real numbers different two by two. Then its edge-labeled graphG[LDES 1

m ] or G[LDE nm ] is shown in Fig.1.1.

B 01 B 0

2

B 03 B 0

4

{eλ 3 t }

{eλ 4 t , eλ 5 t }

{eλ 5 t }

{eλ 3 t , eλ 5 t}{eλ 1 t , eλ 3 t }

Fig. 1.1

If some functions F i (X ), 1 ≤ i ≤ m are non-linear in ( DES 1m ), we can linearize these

non-linear equations X = F i (X ) at the point 0, i.e., if

F i (X ) = F ′i (0)X + R i (X ),

where F ′i (0) is an n ×n matrix, we replace the ith equation X = F i (X ) by a linear differentialequation

X = F ′i (0)X

in (DES 1m ). Whence, we get a uniquely linear differential equation system ( LDES 1

m ) from

(DES 1m ) and its basis graph G[LDES

1m ]. Such a basis graph G[LDES

1m ] of linearized differen-tial equation system ( DES 1

m ) is dened to be the linearized basis graph of (DES 1m ) and denoted

by G[DES 1m ].

All of these notions will contribute to the characterizing of non-solvable differential equationsystems. For terminologies and notations not mentioned here, we follow the [13] for differentialequations, [2] for linear algebra, [3]-[6], [11]-[12] for graphs and Smarandache systems, and [1],[12] for mechanics.




§2. Non-Solvable Linear Ordinary Differential Equations

2.1 Characteristics of Non-Solvable Linear Ordinary Differential Equations

First, we know the following conclusion for non-solvable linear differential equation systems(LDES 1m ) or (LDE n

m ).

Theorem 2.1 The differential equation system (LDES 1m ) is solvable if and only if

(|A1 −λI n × n , |A2 −λI n × n |, · · · , |Am −λI n × n |) = 1

i.e., (LDEq ) is non-solvable if and only if

(|A1 −λI n × n , |A2 −λI n × n |, · · · , |Am −λI n × n |) = 1 .

Similarly, the differential equation system (LDE nm ) is solvable if and only if

(P 1 (λ), P 2(λ), · · · , P m (λ)) = 1 ,

i.e., (LDE nm ) is non-solvable if and only if

(P 1 (λ), P 2(λ), · · · , P m (λ)) = 1 ,

where P i (λ) = λn + a [0]i1 λn − 1 + · · ·+ a [0]

i (n − 1) λ + a [0]in for integers 1 ≤ i ≤ m.

Proof Let λi 1 , λ i 2 , · · · , λ in be the n solutions of equation |Ai − λI n × n | = 0 and B i thesolution basis of ith differential equation in ( LDES 1

m ) or (LDE nm ) for integers 1 ≤ i ≤ m.

Clearly, if ( LDES 1m ) ((LDE n

m )) is solvable, thenm

i=1

B i

=

, i.e.,

m

i=1 {λ

i1, λ

i 2,

· · · , λ

in } =

by Denition 1 .5 and Theorem 1 .2. Choose λ0

m

i=1 {λ i1 , λ i2 , · · · , λin }. Then ( λ − λ0 ) is a

common divisor of these polynomials |A1 −λI n × n , |A2 −λI n × n |, · · · , |Am −λI n × n |. Thus

(|A1 −λI n × n , |A2 −λI n × n |, · · · , |Am −λI n × n |) = 1 .

Conversely, if

(|A1 −λI n × n , |A2 −λI n × n |, · · · , |Am −λI n × n |) = 1 ,

let (λ

−λ01 ), (λ

−λ02 ),

· · · , (λ

−λ0l ) be all the common divisors of polynomials

|A1

−λI n × n ,

|A2

−λI n × n |, · · · , |Am −λI n × n |, where λ0i = λ0j if i = j for 1 ≤ i, j ≤ l. Then it is clear that

C 1eλ 01 + C 2eλ 02 + · · ·+ C l eλ 0 l

is a solution of (LEDq ) ((LDE nm )) for constants C 1 , C 2 , · · · , C l .

For discussing the non-solvable space of a linear differential equation system ( LEDS 1m ) or

(LDE nm ) in details, we introduce the following conception.






bn − 1x + bn with roots x1 , x2 , · · · , xm and y1 , y2 , · · · , yn , respectively. Dene a matrix

V (f, g ) =

a0 a1 · · · am 0 · · · 0 0

0 a0 a1 · · · am 0 · · · 0

· · · · · · · · · · · · · · · · · · · · · · · ·0 · · · 0 0 a0 a1 · · · am

b0 b1 · · · bn 0 · · · 0 0

0 b0 b1 · · · bn 0 · · · 0

· · · · · · · · · · · · · · · · · · · · · · · ·0 · · · 0 0 b0 b1 · · · bn

Then R(f, g ) = det V (f, g ).

We get the following result immediately by Theorem 2 .3.

Corollary 2.5 (1) For two integers 1 ≤ i, j ≤ m, two differential equations (LDES 1ij ) are

parallel in (LDES 1m ) if and only if

R(|Ai −λI n × n |, |Aj −λI n × n |) = 0 ,

particularly, the homogenous equations

V (|Ai −λI n × n |, |Aj −λI n × n |)X = 0

have only solution (0, 0,

· · · , 0

2n

)T if

|Ai

− λI n × n

| = a0λn + a1λn − 1 +

· · · + an − 1λ + an and

|Aj −λI n × n | = b0λn + b1λn − 1 + · · ·+ bn − 1λ + bn .(2) For two integers 1 ≤ i, j ≤ m, two differential equations (LDE n

ij ) are parallel in (LDE n

m ) if and only if R(P i (λ), P j (λ)) = 0 ,

particularly, the homogenous equations V (P i (λ), P j (λ))X = 0 have only solution (0, 0, · · · , 0

2n

)T .

Proof Clearly, |Ai −λI n × n | and |Aj −λI n × n | have no same roots if and only if

R(|Ai −λI n × n |, |Aj −λI n × n |) = 0 ,

which implies that the two differential equations ( LEDS 1ij ) are parallel in ( LEDS 1

m ) and thehomogenous equations

V (|Ai −λI n × n |, |Aj −λI n × n |)X = 0

have only solution (0 , 0, · · · , 0

2n

)T . That is the conclusion (1). The proof for the conclusion (2)

is similar.






be a Jordan black of ki ×ki and

A = T

J 1 O

J 2.. .

O J s

T − 1 .

Then we are easily know the solution basis of the linear differential equation system

dX dt

= AX (LDES 1)

with X = [x1 (t), x2 (t), · · · , xn v (t)]T is nothing but B v by Theorem 1 .2. Notice that the Jordanblack and the matrix T are uniquely determined by B v . Thus the linear homogeneous differen-tial equation ( LDES 1) is uniquely determined by B v . It should be noted that this constructioncan be processed on each vertex v V (G). We nally get a linear homogeneous differentialequation system ( LDES 1

m), which is uniquely determined by the basis graph G.

Similarly, we construct the linear homogeneous differential equation system ( LDE nm ) for

the basis graph G. In fact, for u V (G), let the basis B u at the vertex u be B u = {t leα i t | 1 ≤i ≤ s, 1 ≤ l ≤ ki}. Notice that λi should be a ki -fold zero of the characteristic equation P (λ) = 0with k1 + k2 + · · ·+ ks = n . Thus P (λ i ) = P ′ (λ i ) = · · · = P (k i − 1) (λ i ) = 0 but P (k i ) (λ i ) = 0for integers 1 ≤ i ≤ s. Dene a polynomial P u (λ) following

P u (λ) =s

i =1

(λ −λ i )k i

associated with the vertex u. Let its expansion be

P u (λ) = λn + au 1λn − 1 +

· · ·+ au (n − 1) λ + aun .

Now we construct a linear homogeneous differential equation

x(n ) + au 1x(n − 1) + · · ·+ au (n − 1) x ′ + aun x = 0 (Lh DE n )

associated with the vertex u. Then by Theorem 1 .2 we know that the basis solution of ( LDE n )is just C u . Notices that such a linear homogeneous differential equation ( LDE n ) is uniquelyconstructed. Processing this construction for every vertex u V (G), we get a linear homoge-neous differential equation system ( LDE n

m ). This completes the proof.

Example 2.8 Let (LDE nm ) be the following linear homogeneous differential equation system

x −3x + 2 x = 0 (1)x −5x + 6 x = 0 (2)

x −7x + 12 x = 0 (3)

x −9x + 20 x = 0 (4)

x −11x + 30 x = 0 (5)

x −7x + 6 x = 0 (6)






Example 2.12 Let (LDE nm ) ′ be the following linear homogeneous differential equation system

x + 3 x + 2 x = 0 (1)

x + 5 x + 6 x = 0 (2)

x + 7 x + 12 x = 0 (3)x + 9 x + 20 x = 0 (4)

x + 11 x + 30 x = 0 (5)

x + 7 x + 6 x = 0 (6)

Then its basis graph is shown in Fig.2 .2.Let : H → H ′ be determined by ({eλ i t , eλ j t }) = {e− λ i t , e− λ j t } and

({eλ i t , eλ j t } {eλ k t , eλ l t }) = {e− λ i t , e− λ j t } {e− λ k t , e− λ l t}for integers 1 ≤ i, k ≤ 6 and j = i + 1 ≡ 6(mod6), l = k + 1 ≡ 6(mod6). Then it is clear that

H

H ′ . Thus ( LDE nm ) ′ is combinatorially equivalent to the linear homogeneous differentialequation system ( LDE n

m ) appeared in Example 2 .8.

Denition 2.13 Let G be a simple graph. A vertex-edge labeled graph θ : G → Z + is called integral if θ(uv) ≤ min{θ(u), θ(v)} for uv E (G), denoted by G I θ .

Let GI θ1 and GI τ

2 be two integral labeled graphs. They are called identical if G1 G2 and θ(x) = τ ( (x)) for any graph isomorphism and x V (G1 ) E (G1 ), denoted by GI θ

1 = GI τ 2 .

For example, these labeled graphs shown in Fig.2 .3 are all integral on K 4−e, but GI θ1 = GI τ

2 ,GI θ

1 = GI σ3 .

3 4

4 3

1

2

2

1 2 2 1 1

4 2

2 4

3

3

3 3

4 4

2

GI θ1 GI τ

2

2 2

1

1GI σ

3

Fig. 2.3

Let G[LDES 1m ] (G[LDE n

m ]) be a basis graph of the linear homogeneous differential equa-tion system ( LDES 1

m ) (or (LDE nm )) labeled each v V (G[LDES 1

m ]) (or v V (G[LDE nm ]))

by B v . We are easily get a vertex-edge labeled graph by relabeling v V (G[LDES 1m ]) (orv V (G[LDE n

m ])) by |B v | and uv E (G[LDES 1m ]) (or uv E (G[LDE n

m ])) by |B u B v |.Obviously, such a vertex-edge labeled graph is integral, and denoted by GI [LDES 1

m ] (orGI [LDE n

m ]). The following result completely characterizes combinatorially equivalent linearhomogeneous differential equation systems.

Theorem 2.14 Let (LDES 1m ), (LDES 1

m )′ (or (LDE nm ), (LDE n

m )′ ) be two linear homogeneous



14 Linfan Mao

differential equation systems with integral labeled graphs H, H ′ . Then (LDES 1m )

(LDES 1

m ) ′

(or (LDE nm )

(LDE n

m )′ ) if and only if H = H ′ .

Proof Clearly, H = H ′ if (LDES 1m )

(LDES 1

m )′ (or (LDE nm )

(LDE n

m )′ ) by deni-

tion. We prove the converse, i.e., if H = H ′

then there must be ( LDES 1m )

(LDES 1m )

′

(or(LDE nm )

(LDE n

m )′ ).

Notice that there is an objection between two nite sets S 1 , S 2 if and only if |S 1| = |S 2|.Let τ be a 1 −1 mapping from B v on basis graph G[LDES 1

m ] (or basis graph G[LDE nm ]) to

B v ′ on basis graph G[LDES 1m ]′ (or basis graph G[LDE n

m ]′ ) for v, v′ V (H ′ ). Now if H = H ′ ,we can easily extend the identical isomorphism idH on graph H to a 1 − 1 mapping idH :G[LDES 1

m ] → G[LDES 1m ]′ (or idH : G[LDE n

m ] → G[LDE nm ]′ ) with labelings θ : v →B v and

θ′v ′ : v′ → B v ′ on G[LDES 1

m ], G[LDES 1m ]′ (or basis graphs G[LDE n

m ], G[LDE nm ]′ ). Then

it is an immediately to check that idH θ(x) = θ′ τ (x) for x V (G[LDES 1m ]) E (G[LDES 1

m ])(or for x V (G[LDE n

m ]) E (G[LDE nm ])). Thus idH is an isomorphism between basis graphs

G[LDES 1m ] and G[LDES 1

m ]′ (or G[LDE nm ] and G[LDE n

m ]′ ). Thus ( LDES 1m )

id H (LDES 1m ) ′

(or (LDE nm )

id H (LDE nm )′ ). This completes the proof.

According to Theorem 2 .14, all linear homogeneous differential equation systems ( LDES 1m )

or (LDE nm ) can be classied by G-solutions into the following classes:

Class 1. G[LDES 1m ] K m or G[LDE n

m ] K m for integers m, n ≥ 1.

The G-solutions of differential equation systems are labeled by solution bases on K m andany two linear differential equations in ( LDES 1

m ) or (LDE nm ) are parallel, which characterizes

m isolated systems in this class.

For example, the following differential equation system

x + 3 x + 2 x = 0

x −5x + 6 x = 0

x + 2 x −3x = 0

is of Class 1.

Class 2. G[LDES 1m ] K m or G[LDE n

m ] K m for integers m, n ≥ 1.

The G-solutions of differential equation systems are labeled by solution bases on completegraphs K m in this class. By Corollary 2 .6, we know that G[LDES 1

m ] K m or G[LDE nm ] K m

if (LDES 1m ) or (LDE nm ) is solvable. In fact, this implies that

v V (K m )

B v =u,v V (K m )

(B u B v ) = .

Otherwise, ( LDES 1m ) or (LDE n

m ) is non-solvable.

For example, the underlying graphs of linear differential equation systems (A) and (B) in





16 Linfan Mao

Denition 2.18 Let dX/dt = Av X , x(n ) + av1 x(n − 1) + · · ·+ avn x = 0 be differential equations associated with vertex v and H a spanning subgraph of G[LDES 1

m ] (or G[LDE nm ]). A point

X R n is called a H -equilibrium point if Av X = 0 in (LDES 1m ) with initial value X v (0)

or (X )n + av1(X )n − 1 + ·· · + avn X = 0 in (LDE nm ) with initial values xv (0) , x ′

v (0) , · · · ,x

(n − 1)v (0) for v V (H ).

We consider only two kind of stabilities on the zero G-solution of linear homogeneousdifferential equations in this section. One is the sum-stability. Another is the prod-stability.

2.3.1 Sum-Stability

Denition 2.19 Let H be a spanning subgraph of G[LDES 1m ] or G[LDE n

m ] of the linear homogeneous differential equation systems (LDES 1

m ) with initial value X v (0) or (LDE nm ) with

initial values xv (0) , x ′v (0) , · · · , x(n − 1)

v (0). Then G[LDES 1m ] or G[LDE n

m ] is called sum-stable or asymptotically sum-stable on H if for all solutions Y v (t), v V (H ) of the linear differential equations of (LDES 1

m ) or (LDE nm ) with

|Y v (0)

−X v (0)

| < δ v exists for all t

≥ 0,

| v V (H )

Y v (t)

−v V (H )

X v (t)| < ε , or furthermore, limt → 0 | v V (H )

Y v (t) − v V (H )X v (t)| = 0 .

Clearly, an asymptotic sum-stability implies the sum-stability of that G[LDES 1m ] or G[LDE n

m ].The next result shows the relation of sum-stability with that of classical stability.

Theorem 2.20 For a G-solution G[LDES 1m ] of (LDES 1

m ) with initial value X v (0) (or G[LDE nm ]

of (LDE nm ) with initial values xv (0), x ′

v (0) , · · · , x(n − 1)v (0)), let H be a spanning subgraph of

G[LDES 1m ] (or G[LDE n

m ]) and X an equilibrium point on subgraphs H . If G[LDES 1m ] (or

G[LDE nm ]) is stable on any v V (H ), then G[LDES 1

m ] (or G[LDE nm ]) is sum-stable on H .

Furthermore, if G[LDES 1m ] (or G[LDE n

m ]) is asymptotically sum-stable for at least one vertex

v V (H ), then G[LDES 1m ] (or G[LDE

nm ]) is asymptotically sum-stable on H .

Proof Notice that

|v V (H )

pv Y v (t) −v V (H )

pv X v (t)| ≤v V (H )

pv |Y v (t) −X v (t)|and

limt →0 |

v V (H )

pv Y v (t) −v V (H )

pv X v (t)| ≤v V (H )

pv limt →0 |Y v (t) −X v (t)|.

Then the conclusion on sum-stability follows.

For linear homogenous differential equations ( LDES 1) (or (LDE n )), the following resulton stability of its solution X (t) = 0 (or x(t) = 0) is well-known.

Lemma 2.21 Let γ = max { Reλ| |A −λI n × n | = 0}. Then the stability of the trivial solution X (t) = 0 of linear homogenous differential equations (LDES 1) (or x(t) = 0 of (LDE n )) is determined as follows:

(1) if γ < 0, then it is asymptotically stable;




(2) if γ > 0, then it is unstable;(3) if γ = 0 , then it is not asymptotically stable, and stable if and only if m′ (λ) = m(λ)

for every λ with Reλ = 0 , where m(λ) is the algebraic multiplicity and m ′ (λ) the dimension of eigenspace of λ.

By Theorem 2 .20 and Lemma 2 .21, the following result on the stability of zero G-solutionof (LDES 1

m ) and ( LDE nm ) is obtained.

Theorem 2.22 A zero G-solution of linear homogenous differential equation systems (LDES 1m )

(or (LDE nm )) is asymptotically sum-stable on a spanning subgraph H of G[LDES 1

m ] (or G[LDE nm ])

if and only if Reα v < 0 for each β v (t)eα v t B v in (LDES 1) or Reλv < 0 for each t lv eλ v t C v

in (LDE nm ) hold for v V (H ).

Proof The sufficiency is an immediately conclusion of Theorem 2 .20.Conversely, if there is a vertex v V (H ) such that Re α v ≥ 0 for β v (t)eα v t B v in

(LDES 1) or Reλv

≥ 0 for t lv eλ v t C v in (LDE n

m ), then we are easily knowing that

limt →∞

β v (t)eα v t → ∞if α v > 0 or β v (t) =constant, and

limt →∞

t lv eλ v t → ∞if λv > 0 or lv > 0, which implies that the zero G-solution of linear homogenous differentialequation systems ( LDES 1) or (LDE n ) is not asymptotically sum-stable on H .

The following result of Hurwitz on real number of eigenvalue of a characteristic polynomialis useful for determining the asymptotically stability of the zero G-solution of ( LDES 1

m ) and(LDE n

m ).

Lemma 2.23 Let P (λ) = λn + a1λn − 1 + · · ·+ an − 1λ + an be a polynomial with real coefficients a i , 1 ≤ i ≤ n and

∆ 1 = |a1|, ∆ 2 =a1 1

a3 a2, · · ·∆ n =

a1 1 0 · · · 0

a3 a2 a1 0 · · · 0

a5 a4 a3 a2 a1 0 · · · 0

· · · · · · · · · · · · · · · · · · · · · · · ·0 · · · an

.

Then Reλ < 0 for all roots λ of P (λ) if and only if ∆ i > 0 for integers 1

≤ i

≤ n.

Thus, we get the following result by Theorem 2 .22 and lemma 2 .23.

Corollary 2.24 Let ∆ v1 , ∆ v

2 , · · · , ∆ vn be the associated determinants with characteristic polyno-

mials determined in Lemma 4.8 for v V (G[LDES 1m ]) or V (G[LDE n

m ]). Then for a spanning subgraph H < G [LDES 1

m ] or G[LDE nm ], the zero G-solutions of (LDES 1

m ) and (LDE nm ) is

asymptotically sum-stable on H if ∆ v1 > 0, ∆ v

2 > 0, · · · , ∆ vn > 0 for v V (H ).






(or (LDE nm )) is asymptotically prod-stable if it is asymptotically sum-stable on a spanning

subgraph H of G[LDES 1m ] (or G[LDE n

m ]). Particularly, it is asymptotically prod-stable if the zero solution 0 is stable on v V (H ).

Example 2.29 Let a G-solution of ( LDES 1m ) or (LDE

nm ) be the basis graph shown in Fig.2 .4,where v1 = {e− 2t , e− 3t , e3t}, v2 = {e− 3t , e− 4t}, v3 = {e− 4t , e− 5t , e3t}, v4 = {e− 5t , e− 6t , e− 8t},

v5 = {e− t , e− 6t }, v6 = {e− t , e− 2t , e− 8t}. Then the zero G-solution is sum-stable on the trianglev4v5v6 , but it is not on the triangle v1v2v3 . In fact, it is prod-stable on the triangle v1 v2v3 .

{e− 8t } {e3t }

v1

v2

v3v4

{e− 2t}{e− 3t }

{e− 4t }

{e− 5t

}

{e− 6t }

{e− t}v5

v6

Fig. 2.4 A basis graph

§3. Global Stability of Non-Solvable Non-Linear Differential Equations

For differential equation system ( DES 1m ), we consider the stability of its zero G-solution of

linearized differential equation system ( LDES 1m ) in this section.

3.1 Global Stability of Non-Solvable Differential Equations

Denition 3.1 Let H be a spanning subgraph of G[DES 1m ] of the linearized differential equation

systems (DES 1m ) with initial value X v (0). A point X R n is called a H -equilibrium point of

differential equation system (DES 1m ) if f v (X ) = 0 for v V (H ).

Clearly, 0 is a H -equilibrium point for any spanning subgraph H of G[DES 1m ] by denition.

Whence, its zero G-solution of linearized differential equation system ( LDES 1m ) is a solution

of (DES 1m ).

Denition 3.2 Let H be a spanning subgraph of G[DES 1m ] of the linearized differential equation

systems (DES 1m ) with initial value X v (0) . Then G[DES 1

m ] is called sum-stable or asymptoti-

cally sum-stable on H if for all solutions Y v (t), v V (H ) of (DES 1m ) with Y v (0) −X v (0) < δ v

exists for all t ≥ 0,

v V (H )

Y v (t) −v V (H )

X v (t) < ε,

or furthermore,



20 Linfan Mao

limt →0

v V (H )

Y v (t) −v V (H )

X v (t) = 0 ,

and prod-stable or asymptotically prod-stable on H if for all solutions Y v (t), v

V (H ) of

(DES 1m ) with Y v (0) −X v (0) < δ v exists for all t ≥ 0,

v V (H )

Y v (t) −v V (H )

X v (t) < ε,

or furthermore,

limt →0

v V (H )

Y v (t) −v V (H )

X v (t) = 0 .

Clearly, the asymptotically sum-stability or prod-stability implies respectively that thesum-stability or prod-stability.

Then we get the following result on the sum-stability and prod-stability of the zero G-solution of ( DES 1

m ).

Theorem 3.3 For a G-solution G[DES 1m ] of differential equation systems (DES 1

m ) with initial value X v (0) , let H 1 , H 2 be spanning subgraphs of G[DES 1

m ]. If the zero G-solution of (DES 1m )

is sum-stable or asymptotically sum-stable on H 1 and H 2 , then the zero G-solution of (DES 1m )

is sum-stable or asymptotically sum-stable on H 1 H 2 .Similarly, if the zero G-solution of (DES 1

m ) is prod-stable or asymptotically prod-stable on H 1 and X v (t) is bounded for v V (H 2 ), then the zero G-solution of (DES 1

m ) is prod-stable or asymptotically prod-stable on H 1 H 2 .

Proof Notice that

X 1 + X 2 ≤ X 1 + X 2 and X 1X 2 ≤ X 1 X 2

in R n . We know that

v V (H 1 ) V (H 2 )

X v (t) =v V (H 1 )

X v (t) +v V (H 2 )

X v (t)

≤v V (H 1 )

X v (t) +v V (H 2 )

X v (t)

and

v V (H 1 ) V (H 2 )

X v (t) =v V (H 1 )

X v (t)v V (H 2 )

X v (t)

≤v V (H 1 )

X v (t)v V (H 2 )

X v (t) .




Whence,

v V (H 1 ) V (H 2 )

X v (t) ≤ or limt →0

v V (H 1 ) V (H 2 )

X v (t) = 0

if = 1 + 2 with

v V (H 1 )

X v (t) ≤ 1 andv V (H 2 )

X v (t) ≤ 2

or

limt → 0

v V (H 1 )

X v (t) = 0 and limt → 0

v V (H 2 )

X v (t) = 0 .

This is the conclusion (1). For the conclusion (2), notice that

v V (H 1 ) V (H 2 )

X v (t) ≤v V (H 1 )

X v (t)v V (H 2 )

X v (t) ≤ M

if

v V (H 1 )

X v (t) ≤ andv V (H 2 )

X v (t) ≤ M.

Consequently, the zero G-solution of ( DES 1m ) is prod-stable or asymptotically prod-stable on

H 1 H 2 .

Theorem 3 .3 enables one to get the following conclusion which establishes the relation of stability of differential equations at vertices with that of sum-stability and prod-stability.

Corollary 3.4 For a G-solution G[DES 1m ] of differential equation system (DES 1

m ) with initial value X v (0) , let H be a spanning subgraph of G[DES 1m ]. If the zero solution is stable or asymptotically stable at each vertex v V (H ), then it is sum-stable, or asymptotically sum-stable and if the zero solution is stable or asymptotically stable in a vertex u V (H ) and X v (t)is bounded for v V (H ) \ {u}, then it is prod-stable, or asymptotically prod-stable on H .

It should be noted that the converse of Theorem 3 .3 is not always true. For example, let

v V (H 1 )

X v (t) ≤ a + andv V (H 2 )

X v (t) ≤ −a + .

Then the zero G-solution G[DES 1m ] of differential equation system ( DES 1

m ) is not sum-stable

on subgraphs H 1 and H 2 , but

v V (H 1 H 2 )

X v (t) ≤v V (H 1 )

X v (t) +v V (H 2 )

X v (t) = .

Thus the zero G-solution G[DES 1m ] of differential equation system ( DES 1

m ) is sum-stable onsubgraphs H 1 H 2 . Similarly, let



22 Linfan Mao

v V (H 1 )

X v (t) ≤ t r and

v V (H 2 )

X v (t) ≤ tr

for a real number r. Then the zero G-solution G[DES 1m ] of (DES 1m ) is not prod-stable onsubgraphs H 1 and X v (t) is not bounded for v V (H 2) if r > 0. However, it is prod-stable onsubgraphs H 1 H 2 for

v V (H 1 H 2 )

X v (t) ≤v V (H 1 )

X v (t)v V (H 2 )

X v (t) = .

3.2 Linearized Differential Equations

Applying these conclusions on linear differential equation systems in the previous section, wecan nd conditions on F i (X ), 1 ≤ i ≤ m for the sum-stability and prod-stability at 0 following.For this objective, we need the following useful result.

Lemma 3.5([13]) Let X = AX + B(X ) be a non-linear differential equation, where A is a constant n ×n matrix and Reλ i < 0 for all eigenvalues λ i of A and B (X ) is continuous dened on t ≥ 0, X ≤ α with

limX → 0

B (X )X

= 0 .

Then there exist constants c > 0, β > 0 and δ, 0 < δ < α such that

X (0) ≤ ε ≤ δ 2c

implies that X (t) ≤ cεe− βt/ 2 .

Theorem 3.6 Let (DES 1m ) be a non-linear differential equation system, H a spanning subgraph

of G[DES 1m ] and

F v (X ) = F ′v 0 X + Rv (X )

such that

limX → 0

Rv (X )X

= 0

for v V (H ). Then the zero G-solution of (DES 1m ) is asymptotically sum-stable or asymp-

totically prod-stable on H if Reα v < 0 for each β v (t)eα v t B v , v V (H ) in (DES 1m ).

Proof Dene c = max {cv , v V (H )}, ε = min {εv , v V (H )} and β = min {β v , v V (H )}. Applying Lemma 3 .5, we know that for v V (H ),

X v (0) ≤ ε ≤ δ 2c

implies that X v (t) ≤ cεe− βt/ 2 .








But then there is a contradiction

L v V (H )

X v (tn + η) < L (Y )

yields by letting Y v (0) = v V (H ) X v (tn ) for sufficiently large n. Thus, there must be Y v = X .Whence, the zero G-solution G[DES 1

m ] is asymptotically sum-stable on H by denition. Thisis the conclusion (1).

Similarly, we can prove the conclusion (2).

The following result shows the combination of Liapunov sum-functions or prod-functions.

Theorem 3.9 For a G-solution G[DES 1m ] of a differential equation system (DES 1

m ) with initial value X v (0), let H 1 , H 2 be spanning subgraphs of G[DES 1

m ], X an equilibrium point of (DES 1

m ) on H 1 H 2 and R+ (x, y ) =

i ≥ 0,j ≥ 0

a i,j x i yj

be a polynomial with a i,j ≥ 0 for integers i, j ≥ 0. Then R+ (L1 , L2) is a Liapunov sum-function or Liapunov prod-function on X for H 1 H 2 with conventions for integers i, j, k,l ≥ 0 that

a ij L i1L j

2 v V (H 1 V (H 2 )

X v (t) + akl Lk1 L l

2 v V (H 1 V (H 2 )

X v (t)

= aij L i1 v V (H 1 )

X v (t) L j2 v V (H 2 )

X v (t)

+ akl Lk1

v V (H

1)

X v (t) L l2

v V (H

2)

X v (t)

if L1 , L2 are Liapunov sum-functions and

a ij L i1L j

2 v V (H 1 V (H 2 )


2 v V (H 1 V (H 2 )

X v (t)

= aij L i1 v V (H 1 )

X v (t) L j2 v V (H 2 )

X v (t)

+ akl Lk1 v V (H 1 )

X v (t) L l2 v V (H 2 )

X v (t)

if L1 , L2 are Liapunov prod-functions on X for H 1 and H 2 , respectively. Particularly, if there is a Liapunov sum-function (Liapunov prod-function) L on H 1 and H 2 , then L is also a Liapunov sum-function (Liapunov prod-function) on H 1 H 2 .

Proof Notice that

d a ij L i1L j

2

dt = aij iL i − 1

1 L1L j

2 + jL i1L j − 1

1 L2



26 Linfan Mao

if i, j ≥ 1. Whence,

a ij L i1L j

2 v V (H 1 V (H 2 )

X v (t) ≥ 0

if

L1 v V (H 1 )

X v (t) ≥ 0 and L2 v V (H 2 )

X v (t) ≥ 0

andd(a ij L i

1L j2)

dt v V (H 1 V (H 2 )

X v (t) ≤ 0

if

L1 v V (H 1 )

X v (t) ≤ 0 and L2 v V (H 2 )

X v (t) ≤ 0.

Thus R+ (L1 , L2) is a Liapunov sum-function on X for H 1 H 2 .Similarly, we can know that R+ (L1 , L2) is a Liapunov prod-function on X for H 1 H 2 if

L1 , L2 are Liapunov prod-functions on X for H 1 and H 2 .

Theorem 3 .9 enables one easily to get the stability of the zero G-solutions of ( DES 1m ).

Corollary 3.10 For a differential equation system (DES 1m ), let H < G [DES 1

m ] be a spanning subgraph. If Lv is a Liapunov function on vertex v for v V (H ), then the functions

LH S =

v V (H )

Lv and L H P =

v V (H )

Lv

are respectively Liapunov sum-function and Liapunov prod-function on graph H . Particularly,

if L = Lv for v V (H ), then L is both a Liapunov sum-function and a Liapunov prod-function on H .

Example 3.11 Let (DES 1m ) be determined by

dx1 /dt = λ11 x1

dx2 /dt = λ12 x2

· · · · · · · · ·dxn /dt = λ1n xn

dx1 /dt = λ21 x1

dx2 /dt = λ22 x2

· · · · · · · · ·dxn /dt = λ2n xn

· · ·

dx1 /dt = λn 1x1

dx2 /dt = λn 2x2

· · · · · · · · ·dxn /dt = λnn xn

where all λij , 1 ≤ i ≤ m, 1 ≤ j ≤ n are real and λ ij 1 = λ ij 2 if j 1 = j 2 for integers 1 ≤ i ≤ m.Let L = x2

1 + x2

2 +

· · ·+ x2

n. Then

L = λ i 1x21 + λ i 2x2

2 + · · ·+ λ in x2n

for integers 1 ≤ i ≤ n. Whence, it is a Liapunov function for the ith differential equation if λ ij < 0 for integers 1 ≤ j ≤ n. Now let H < G [LDES 1

m ] be a spanning subgraph of G[LDES 1m ].

Then L is both a Liapunov sum-function and a Liapunov prod-function on H if λvj < 0 forv V (H ) by Corollaries 3 .10.




Theorem 3.12 Let L : O → R be a differentiable function with L(0) = 0 and L v V (H )X >

0 always holds in an area of its -neighborhood U ( ) of 0 for ε > 0, denoted by U + (0, ε) such

area of ε-neighborhood of 0 with L

v V (H )

X > 0 and H < G [DES 1m ] be a spanning subgraph.

(1) If

L v V (H )

X ≤ M

with M a positive number and

L v V (H )

X > 0

in U + (0, ), and for > 0, there exists a positive number c1 , c2 such that

L v V (H )

X ≥ c1 > 0 implies L v V (H )

X ≥ c2 > 0,

then the zero G-solution G[DES 1m ] is not sum-stable on H . Such a function L : O → R is

called a non-Liapunov sum-function on H .(2) If

L v V (H )

X ≤ N

with N a positive number and

L

v V (H )

X > 0

in U + (0, ), and for > 0, there exists positive numbers d1 , d2 such that

L v V (H )

X ≥ d1 > 0 implies L v V (H )

X ≥ d2 > 0,

then the zero G-solution G[DES 1m ] is not prod-stable on H . Such a function L : O → R is

called a non-Liapunov prod-function on H .

Proof Generally, if L(X ) is bounded and L (X ) > 0 in U + (0, ), and for > 0, thereexists positive numbers c1 , c2 such that if L (X ) ≥ c1 > 0, then L (X ) ≥ c2 > 0, we prove that

there exists t1 > t 0 such that X (t1 , t0) > 0 for a number 0 > 0, where X (t1 , t0) denotesthe solution of ( DES nm ) passing through X (t0). Otherwise, there must be X (t1 , t 0) < 0 for

t ≥ t0 . By L (X ) > 0 we know that L(X (t)) > L (X (t0)) > 0 for t ≥ t0 . Combining this factwith the condition L (X ) ≥ c2 > 0, we get that

L(X (t)) = L(X (t0)) +t

t 0

dL(X (s))ds ≥ L(X (t0)) + c2(t −t0 ).



28 Linfan Mao

Thus L(X (t)) → +∞ if t → +∞, a contradiction to the assumption that L(X ) is bounded.Whence, there exists t1 > t 0 such that

X (t1 , t 0) > 0 .

Applying this conclusion, we immediately know that the zero G-solution G[DES 1m ] is not sum-stable or prod-stable on H by conditions in (1) or (2).

Similar to Theorem 3 .9, we know results for non-Liapunov sum-function or prod-functionby Theorem 3 .12 following.

Theorem 3.13 For a G-solution G[DES 1m ] of a differential equation system (DES 1

m ) with initial value X v (0) , let H 1 , H 2 be spanning subgraphs of G[DES 1

m ], 0 an equilibrium point of (DES 1

m ) on H 1 H 2 . Then R+ (L1 , L2) is a non-Liapunov sum-function or non-Liapunov prod-function on 0 for H 1 H 2 with conventions for

a ij L i1L j2 v V (H 1 V (H 2 )X v (t) + akl Lk1 L l2 v V (H 1 V (H 2 )

X v (t)

and

a ij L i1L j

2 v V (H 1 V (H 2 )


2 v V (H 1 V (H 2 )

X v (t)

the same as in Theorem 3.9 if L1 , L2 are non-Liapunov sum-functions or non-Liapunov prod- functions on 0 for H 1 and H 2 , respectively. Particularly, if there is a non-Liapunov sum- function (non-Liapunov prod-function) L on H 1 and H 2 , then L is also a non-Liapunov sum- function (non-Liapunov prod-function) on H 1 H 2 .

Proof Similarly, we can show that R+

(L1 , L2) satises these conditions on H 1 H 2 fornon-Liapunov sum-functions or non-Liapunov prod-functions in Theorem 3 .12 if L1 , L2 arenon-Liapunov sum-functions or non-Liapunov prod-functions on 0 for H 1 and H 2 , respectively.Thus R+ (L1 , L2) is a non-Liapunov sum-function or non-Liapunov prod-function on 0.

Corollary 3.14 For a differential equation system (DES 1m ), let H < G [DES 1

m ] be a spanning subgraph. If Lv is a non-Liapunov function on vertex v for v V (H ), then the functions

LH S =

v V (H )

Lv and L H P =

v V (H )

Lv

are respectively non-Liapunov sum-function and non-Liapunov prod-function on graph H . Par-

ticularly, if L = Lv for v V (H ), then L is both a non-Liapunov sum-function and a non-Liapunov prod-function on H .

Example 3.15 Let (DES 1m ) be

x1 = λ1x21 −λ1 x2

2

x2 = λ1

2 x1 x2

x2 = λ2 x21 −λ2x2

2

x2 = λ2

2 x1 x2

· · ·x1 = λm x2

1 −λm x22

x2 = λm

2 x1 x2




with constants λi > 0 for integers 1 ≤ i ≤ m and L(x1 , x2) = x21 − 2x2

2 . Then L(x1 , x2) =4λ i x1L(x1 , x2) for the i-th equation in ( DES 1

m ). Calculation shows that L(x1 , x2) > 0 if x1 > √ 2x2 or x1 < −√ 2x2 and L(x1 , x2) > 4c

32 for L(x1 , x2) > c in the area of L(x1 , x2) > 0.

Applying Theorem 3 .12, we know the zero solution of ( DES 1m ) is not stable for the i-th equation

for any integer 1 ≤ i ≤ m. Applying Corollary 3 .14, we know that L is a non-Liapunov sum-function and non-Liapunov prod-function on any spanning subgraph H < G [DES 1

m ].

§4. Global Stability of Shifted Non-Solvable Differential Equations

The differential equation systems ( DES 1m ) discussed in previous sections are all in a same

Euclidean space R n . We consider the case that they are not in a same space R n , i.e., shifteddifferential equation systems in this section. These differential equation systems and theirnon-solvability are dened in the following.

Denition 4.1 A shifted differential equation system (SDES 1

m) is such a differential equation

system X 1 = F 1(X 1), X 2 = F 2(X 2), · · · , ˙X m = F m (X m ) (SDES 1

m )

with X 1 = ( x1 , x2 , · · · , xl , x1( l+1) , x1( l+2) , · · · , x1n ),

X 2 = ( x1 , x2 , · · · , xl , x2( l+1) , x2( l+2) , · · · , x2n ),

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,

X m = ( x1 , x2 , · · · , xl , xm (l+1) , xm (l+2) , · · · , xmn ),

where x1 , x2 , · · · , xl , x i ( l+ j ) , 1 ≤ i ≤ m, 1 ≤ j ≤ n −l are distinct variables and F s : R n → R n

is continuous such that F s (0) = 0 for integers 1 ≤ s ≤ m.

A shifted differential equation system (SDES 1m ) is non-solvable if there are integers i,j, 1 ≤i, j ≤ m and an integer k, 1 ≤ k ≤ l such that x[i ]

k (t) = x[j ]k (t), where x[i ]

k (t), x [j ]k (t) are solutions

xk (t) of the i-th and j -th equations in (SDES 1m ), respectively.

The number dim( SDES 1m ) of variables x1 , x2 , · · · , xl , x i ( l+ j ) , 1 ≤ i ≤ m, 1 ≤ j ≤ n −l in

Denition 4 .1 is uniquely determined by ( SDES 1m ), i.e., dim( SDES 1

m ) = mn −(m −1)l. Forclassifying and nding the stability of these differential equations, we similarly introduce thelinearized basis graphs G[SDES 1

m ] of a shifted differential equation system to that of ( DES 1m ),

i.e., a vertex-edge labeled graph with

V (G[SDES 1m ]) = {B i |1 ≤ i ≤ m},

E (G[SDES 1m ]) =

{(B i , B j )

|B i B j

=

, 1

≤ i, j

≤ m

},

where B i is the solution basis of the i-th linearized differential equation X i = F ′i (0)X i forintegers 1 ≤ i ≤ m, called such a vertex-edge labeled graph G[SDES 1

m ] the G-solution of (SDES 1

m ) and its zero G-solution replaced B i by (0, · · · , 0) (|B i | times) and B i B j by(0, · · · , 0) (|B i B j | times) for integers 1 ≤ i, j ≤ m.

Let ( LDES 1m ), (LDES 1

m )′ be linearized differential equation systems of shifted differentialequation systems ( SDES 1

m ) and ( SDES 1m ) with G-solutions H, H ′ . Similarly, they are called



30 Linfan Mao

combinatorially equivalent if there is an isomorphism : H → H ′ of graph and labelingsθ, τ on H and H ′ respectively such that θ (x) = τ (x) for x V (H ) E (H ), denoted by(SDES 1

m )

(SDES 1m )′ . Notice that if we remove these superuous variables from G[SDES 1

m ],then we get nothing but the same vertex-edge labeled graph of ( LDES 1

m ) in R l . Thus we can

classify shifted differential similarly to ( LDES 1m ) in R

l. The following result can be proved

similarly to Theorem 2 .14.

Theorem 4.2 Let (LDES 1m ), (LDES 1

m )′ be linearized differential equation systems of twoshifted differential equation systems (SDES 1

m ), (SDES 1m )′ with integral labeled graphs H, H ′ .

Then (SDES 1m )

(SDES 1

m )′ if and only if H = H ′ .

The stability of these shifted differential equation systems ( SDES 1m ) is also similarly to

that of ( DES 1m ). For example, we know the results on the stability of ( SDES 1

m ) similar toTheorems 2 .22, 2.27 and 3.6 following.

Theorem 4.3 Let (LDES 1m ) be a shifted linear differential equation systems and H < G [LDES 1

m ]

a spanning subgraph. A zero G-solution of (LDES 1m ) is asymptotically sum-stable on H if and only if Reα v < 0 for each β v (t)eα v t B v in (LDES 1) hold for v V (H ) and it is asymptot-ically prod-stable on H if and only if v V (H )

Reα v < 0 for each β v (t)eα v t B v in (LDES 1).

Theorem 4.4 Let (SDES 1m ) be a shifted differential equation system, H < G [SDES 1

m ] a spanning subgraph and

F v (X ) = F ′v 0 X + Rv (X )

such that lim

X → 0

Rv (X )X

= 0

for

v

V (H ). Then the zero G-solution of (SDES 1

m) is asymptotically sum-stable or asymp-

totically prod-stable on H if Reα v < 0 for each β v (t)eα v t B v , v V (H ) in (SDES 1m ).

For the Liapunov sum-function or Liapunov prod-function of a shifted differential equationsystem ( SDES 1

m ), we choose it to be a differentiable function L : O R dim( SDES 1m ) → R with

conditions in Denition 3 .7 hold. Then we know the following result similar to Theorem 3 .8.

Theorem 4.5 For a G-solution G[SDES 1m ] of a shifted differential equation system (SDES 1

m )with initial value X v (0) , let H be a spanning subgraph of G[DES 1

m ] and X an equilibrium point of (SDES 1

m ) on H .

(1) If there is a Liapunov sum-function L : O R dim( SDES 1m ) → R on X , then the zero

G-solution G[SDES 1m ] is sum-stable on X for H , and furthermore, if

L v V (H )

X v (t) < 0

for v V (H )X v (t) = X , then the zero G-solution G[SDES 1

m ] is asymptotically sum-stable on

X for H .




(2) If there is a Liapunov prod-function L : O R dim( SDES 1m ) → R on X for H , then

the zero G-solution G[SDES 1m ] is prod-stable on X for H , and furthermore, if

L

v V (H )

X v (t) < 0

for v V (H )X v (t) = X , then the zero G-solution G[SDES 1

m ] is asymptotically prod-stable on

X for H .

§5. Applications

5.1 Global Control of Infectious Diseases

An immediate application of non-solvable differential equations is the globally control of infec-tious diseases with more than one infectious virus in an area. Assume that there are three kindgroups in persons at time t, i.e., infected I (t), susceptible S (t) and recovered R(t), and thetotal population is constant in that area. We consider two cases of virus for infectious diseases:

Case 1 There are m known virus V 1 , V 2 , · · · , V m with infected rate ki , heal rate h i for integers 1 ≤ i ≤ m and an person infected a virus V i will never infects other viruses V j for j = i.

Case 2 There are m varying V 1 , V 2 , · · · , V m from a virus V with infected rate ki , heal rate hi

for integers 1 ≤ i ≤ m such as those shown in Fig. 5.1.

V 1 V 2 V m

Fig. 5.1

We are easily to establish a non-solvable differential model for the spread of infectiousviruses by applying the SIR model of one infectious disease following:

S = −k1SI

I = k1SI −h1I

R = h1I

S = −k2SI

I = k2SI −h2I

R = h2I · · ·

S = −km SI

I = km SI −hm I

R = hm I

(DES 1m )

Notice that the total population is constant by assumption, i.e., S + I + R is constant.Thus we only need to consider the following simplied system

S = −k1 SI

I = k1SI −h1 I

S = −k2SI

I = k2SI −h2I · · ·S = −km SI

I = km SI −hm I (DES 1

m )

The equilibrium points of this system are I = 0, the S -axis with linearization at equilibrium



32 Linfan Mao

points

S = −k1S

I = k1S −h1

S = −k2S

I = k2S −h2· · ·

S = −km S

I = km S −hm(LDES 1

m )

Calculation shows that the eigenvalues of the ith equation are 0 and ki S −h i , which is negative,i.e., stable if 0 < S < h i /k i for integers 1 ≤ i ≤ m. For any spanning subgraph H < G [LDES 1

m ],we know that its zero G-solution is asymptotically sum-stable on H if 0 < S < h v /k v forv V (H ) by Theorem 2 .22, and it is asymptotically sum-stable on H if

v V (H )

(kv S −hv ) < 0 i.e., 0 < S <v V (H )

hv v V (H )

kv

by Theorem 2 .27. Notice that if I i (t), S i (t) are probability functions for infectious virusesV i , 1 ≤ i ≤ m in an area, then

m

i =1I i (t) and

m

i =1S i (t) are just the probability functions for

all these infectious viruses. This fact enables one to get the conclusion following for globally

control of infectious diseases.

Conclusion 5.1 For m infectious viruses V 1 , V 2, · · · , V m in an area with infected rate ki , heal rate hi for integers 1 ≤ i ≤ m, then they decline to 0 nally if

0 < S <m

i =1h i

m

i =1ki ,

i.e., these infectious viruses are globally controlled. Particularly, they are globally controlled if each of them is controlled in this area.

5.2 Dynamical Equations of Instable Structure

There are two kind of engineering structures, i.e., stable and instable. An engineering structureis instable if its state moving further away and the equilibrium is upset after being movedslightly. For example, the structure (a) is engineering stable but (b) is not shown in Fig.5 .2,

A1

B1 C 1

A2

B2

C 2

D2

(a) (b)

Fig. 5.2

where each edge is a rigid body and each vertex denotes a hinged connection. The motion of a stable structure can be characterized similarly as a rigid body. But such a way can not beapplied for instable structures for their internal deformations such as those shown in Fig.5 .3.




A B

C D

BA

C D

moves

Fig. 5.3

Furthermore, let P 1 , P 2 , · · · , P m be m particles in R 3 with some relations, for instance,the gravitation between particles P i and P j for 1 ≤ i, j ≤ m. Thus we get an instable structureunderlying a graph G with

V (G) =

{P 1 , P 2 ,

· · · , P m

};

E (G) = {(P i , P j )|there exists a relation between P i and P j }.

For example, the underlying graph in Fig.5 .4 is C 4 . Assume the dynamical behavior of particleP i at time t has been completely characterized by the differential equations X = F i (X, t ),where X = ( x1 , x2 , x3). Then we get a non-solvable differential equation system

X = F i (X, t ), 1 ≤ i ≤ m

underlying the graph G. Particularly, if all differential equations are autonomous, i.e., dependon X alone, not on time t , we get a non-solvable autonomous differential equation system

X = F i (X ), 1

≤ i

≤ m.

All of these differential equation systems particularly answer a question presented in [3] forestablishing the graph dynamics, and if they satisfy conditions in Theorems 2 .22, 2.27 or 3.6,then they are sum-stable or prod-stable. For example, let the motion equations of 4 membersin Fig.5 .3 be respectively

AB : X AB = 0; CD : X CD = 0 , AC : X AC = aAC , BC : X BC = aBC ,

where X AB , X CD , X AC and X BC denote central positions of members AB,CD, AC,BC andaAC , a BC are constants. Solving these equations enable one to get

X AB = cAB t + dAB , X AC = aAC t2 + cAC t + dAC ,X CD = cCD t + dCD , X BC = aBC t2 + cBC t + dBC ,

where cAB , cAC , cCD , cBC , dAB , dAC , dCD , dBC are constants. Thus we get a non-solvable dif-ferential equation system

X = 0; X = 0 , X = aAC , X = aBC ,










[6] Linfan Mao, Combinatorial Geometry with Applications to Field Theory (Second edition),Graduate Textbook in Mathematics, The Education Publisher Inc. 2011.

[7] Linfan Mao, Non-solvable spaces of linear equation systems, International J.Math. Com-bin., Vol.2 (2012), 9-23.

[8] W.S.Massey, Algebraic Topology: An Introduction , Springer-Verlag, New York, etc., 1977.[9] Don Mittleman and Don Jezewski, An analytic solution to the classical two-body problem

with drag, Celestial Mechanics , 28(1982), 401-413.[10] F.Smarandache, Mixed noneuclidean geometries, Eprint arXiv: math/0010119 , 10/2000.[11] F.Smarandache, A Unifying Field in Logics–Neutrosopy: Neturosophic Probability, Set,

and Logic , American research Press, Rehoboth, 1999.[12] Walter Thirring, Classical Mathematical Physics , Springer-Verlag New York, Inc., 1997.[13] Wolfgang Walter, Ordinary Differential Equations , Springer-Verlag New York, Inc., 1998.




mth -Root Randers Change of a Finsler Metric

V.K.Chaubey and T.N.Pandey

Department of Mathematics and Statistics

D.D.U. Gorakhpur University, Gorakhpur (U.P.)-273009, India

E-mail: [email protected], [email protected]

Abstract : In this paper, we introduce a m th -root Randers changed Finsler metric as

L (x, y ) = L (x, y ) + β (x, y ),

where L = {a i 1 i 2 ··· i m (x )y i 1 y i 2 · · · y i m } 1m is a m th -root metric and β -is one form. Further

we obtained the relation between the v- and hv- curvature tensor of m th -root Finsler spaceand its m th -root Randers changed Finsler space and obtained some theorems for its S3 andS4-likeness of Finsler spaces and when this changed Finsler space will be Berwald space(resp. Landsberg space). Also we obtain T-tensor for the m th -root Randers changed Finslerspace F n .

Key Words : Randers change, m th -root metric, Berwald space, Landsberg space, S3 andS4-like Finsler space.

AMS(2010) : 53B40, 53C60

§1. Introduction

Let F n = ( M n , L) be a n-dimensional Finsler space, whose M n is the n-dimensional differen-tiable manifold and L(x, y ) is the Finsler fundamental function. In general, L(x, y ) is a functionof point x = ( x i ) and element of support y = ( yi ), and positively homogeneous of degree onein y. In the year 1971 Matsumoto [6] introduced the transformations of Finsler metric given by

L′

(x, y ) = L(x, y ) + β (x, y )

L′′ 2(x, y ) = L2(x, y ) + β 2 (x, y),

where, β = bi (x)yi is a one-form [1] and bi (x) are components of covariant vector which is afunction of position alone. If L(x, y ) is a Riemannian metric, then the Finsler space with ametric L(x, y) = α(x, y ) + β (x, y ) is known as Randers space which is introduced by G.Randers[5]. In papers [3, 7, 8, 9], Randers spaces have been studied from a geometrical viewpoint andvarious theorem were obtained. In 1978, Numata [10] introduced another β -change of Finslermetric given by L(x, y ) = µ(x, y ) + β (x, y ) where µ = {a ij (y)yi yj }

12 is a Minkowski metric

and β as above. This metric is of similar form of Randers one, but there are different tensor1 Received October 25, 2012. Accepted March 4, 2013.



m th -Root Randers Change of a Finsler Metric 39

properties, because the Riemannian space with the metric α is characterized by C ijk = 0 andon the other hand the locally Minkowski space with the metric µ by Rhijk = 0 , C hij | k = 0.

In the year 1979, Shimada [4] introduced the concept of m th root metric and developed it asan interesting example of Finsler metrics, immediately following M.Matsumoto and S.Numatas

theory of cubic metrics [2]. By introducing the regularity of the metric various fundamentalquantities as a Finsler metric could be found. In particular, the Cartan connection of a Finslerspace with m-th root metric could be discussed from the theoretical standpoint. In 1992-1993,the m-th root metrics have begun to be applied to theoretical physics [11, 12], but the resultsof the investigations are not yet ready for acceding to the demands of various applications.

In the present paper we introduce a m th -root Randers changed Finsler metric as

L(x, y ) = L(x, y) + β (x, y )

where L = {a i 1 i 2 ··· i m (x)yi 1 yi 2 ··· y i m

}1

m is a m th -root metric. This metric is of the similar form tothe Randers one in the sense that the Riemannian metric is replaced with the m th -root metric,due to this we call this change as m th -root Randers change of the Finsler metric. Further weobtained the relation between the v-and hv-curvature tensor of m th -root Finsler space and itsm th -root Randers changed Finsler space and obtained some theorems for its S3 and S4-likenessof Finsler spaces and when this changed Finsler space will be Berwald space (resp. Landsbergspace). Also we obtain T-tensor for the m th -root Randers changed Finsler space F n .

§2. The Fundamental Tensors of F n

We consider an n-dimensional Finsler space F n with a metric L(x, y ) given by

L(x, y ) = L(x, y ) + bi (x)yi (1)

where

L = {a i 1 i 2 ··· i m (x)yi 1 yi 2 · · ·yi m }1

m (2)

By putting

(I) . Lm − 1a i (x, y) = aii 2 ··· i m (x)yi 2 yi 3 · · ·yi m (3)

(II) . Lm − 2a ij (x, y ) = aiji 3 i 4 ··· i m (x)yi 3 yi 4 · · ·yi m

(III) . Lm − 3a ijk (x, y) = a ijki 4 i 5 ··· i m (x)yi 4 yi 5 · · ·yi m

Now differentiating equation (1) with respect to yi , we get the normalized supporting element

li = ∂ i L asli = ai + bi (4)

where ai = li is the normalized supporting element for the mth -root metric. Again differen-tiating above equation with respect to yj , the angular metric tensor h ij = L ∂ i ∂ j L is givenas

h ij

L =

hij

L (5)






To nd the v-curvature tensor of F n , we rst nd (h)hv-torsion tensor C ijk = gir C jrk

C ijk = 1m −1

C ijk + 1

2(m −1)L(h i

j mk + h ik m j + h jk m i ) − (13)

a i

L(1 + q ) {m j mk + 1

(m −1)(m −2) h jk }− 1

(m −1)(1 + q ) ai

C jrk br

where LC ijk = LC jrk a ir = ( m −1){a ijk −(δ ij ak + δ ik a j + a i a jk ) + 2 a i a j ak},

h ij = hjr a ir = ( m −1)(δ ij −a i a j ) (14)

m i = m r a ir = bi −qai , and a ijk = a ir a jrk

From (12) and (14), we have the following identities

C ijr h r p = C rij h pr = ( m −1)C ijp , C ijr m r = C ijr br , (15)

m r hri = ( m −1)m i , m i m i = ( b2 −q 2),

h ir h rj = ( m −1)h ij , hir m r = ( m −1)m i

From (9) and (13), we get after applying the identities (15)

C ijr C rhk = τ (m −1)

C ijr C rhk + 12L

(C ijh mk + C ijk mh + C hjk m i + C hik m j ) (16)

+ 1

2(m −1)(C ijr hhk + C hrk h ij )br +

14(m −1)LL

(b2 −q 2)h ij hhk

+ 14LL

(2h ij mh mk + 2 hkh m i m j + h jh m i mk

+ h jk m i mh + h ih m j mk + h ik m j mh )

Now we shall nd the v-curvature tensor S hijk = C ijr C rhk − C ikr C rhj . The tensor is obtainedfrom (16) and given by

S hijk = Θ ( jk ){ τ

m −1C ijr C rhk + h ij mhk + hhk m ij } (17)

= τ (m −1)

S hijk + Θ ( jk ){h ij mhk + hhk m ij }where

m ij = 1

2(m −1)L {C ijr br + (b2 −q 2)

4L hij +

(m −1)2

L− 1m i m j } (18)

and the symbol Θ ( jk ){···} denotes the exchange of j, k and subtraction.

Proposition 3.1 The v-curvature tensor S hijk of m th -root Randers changed Finsler space F n

with respect to Cartan’s connection C Γ is of the form (17).

It is well known [13] that the v-curvature tensor of any three-dimensional Finsler space isof the form

L2S hijk = S (hhj h ik −hhk h ij ) (19)

Owing to this fact M. Matsumoto [13] dened the S3-like Finsler space F n (n ≥ 3) as such aFinsler space in which v-curvature tensor is of the form (19). The scalar S in (19) is a functionof x alone.



42 V.K.Chaubey and T.N.Pandey

The v-curvature tensor of any four-dimensional Finsler space may be written as [13]

L2S hijk = Θ ( jk ){hhj K ki + h ik K hj } (20)

where K ij is a (0, 2) type symmetric Finsler tensor eld which is such that K ij yj = 0. A Finsler

space F n (n ≥ 4) is called S4-like Finsler space [13] if its v-curvature tensor is of the form (20).From (17), (19), (20) and (5) we have the following theorems.

Theorem 3.1 The m th -root Randers changed S3-like or S4-like Finsler space is S4-like Finsler space.

Theorem 3.2 If v-curvature tensor of mth -root Randers changed Finsler space F n vanishes,then the Finsler space with mth -root metric F n is S4-like Finsler space.

If v-curvature tensor of Finsler space with m th -root metric F n vanishes then equation (17)reduces to

S hijk = hij mhk + hhk m ij

−h ik mhj

−hhj m ik (21)

By virtue of (21) and (11) and the Ricci tensor S ik = ghk S hijk is of the form

S ik = (− 1

(m −1)τ ){mh ik + ( m −1)(n −3)m ik },

where m = m ij a ij , which in view of (18) may be written as

S ik + H 1h ik + H 2C ikr br = H 3m i mk , (22)

where

H 1 = m

(m −1)τ +

(n −3)(b2 −q 2)8(m −1)L2 ,

H 2 = (n −3)2(m −1)L

,

H 3 = −(n −3)

2L2 .

From (22), we have the following

Theorem 3.3 If v-curvature tensor of mth -root Randers changed Finsler space F n vanishes then there exist scalar H 1 and H 2 in Finsler space with mth -root metric F n (n ≥ 4) such that matrix S ik + H 1h ik + H 2C ikr br is of rank two.

§4. The (v)hv-Torsion Tensor and hv-Curvature Tensor of ¯F

n

Now we concerned with ( v)hv-torsion tensor P ijk and hv-curvature tensor P hijk . With respectto the Cartan connection C Γ, L| i = 0 , li | j = 0 , hij | k = 0 hold good [13].

Taking h-covariant derivative of equation (9) and using (4) and li = ai = 0 we have

C ijk | h = τ C ijk | h + bi | h

L C ijk +

(h ij bk | h + h jk bi | h + hki bj | h )2L

(23)





44 V.K.Chaubey and T.N.Pandey

Now using (29), the v-derivative of C ijk is given as

L C ijk |h = τ (Lbh −βl h )

L C ijk + LτC ijk |h −τ

12L

(h ih lj mk + h jh li mk (30)

+ h jh lk m i + hkh lj m i + h ih lk m j + hkh li m j + h ij lh mk

+ h jk lh m i + hki lh m j ) + τ 2

(h ij mk |h + h jk m i |h + hki m j |h )

Using (4), (9) and (30), the T-tensor for m th -root Randers changed Finsler space F n is givenby

T hijk = τ (T hijk + Bhijk ) + τ 2L

(h jk mh li + h ik mh lj + h ij mh lk (31)

+ hki m j lh ) + 12L

(hhj mk bi + h jk mh bi + hkh m j bi + h ik mh bj

+ h ih mk bj + hkh m i bj + h ij mh bk + h ih m j bk + h jh m i bk + h ij mk bh

+ hki m j bh + h jk m i bh ) + τ 2

(h ij mk |h + h jk m i |h + hki m j |h )

+ τ (Lb

h −βl

h)

L C ijk

where Bhijk = βC hij |k + bi C hjk + bj C hik + bk C hij + bh C ijk . Thus, we know

Proposition 5.1 The T-tensor T hijk for mth -root Randers changed Finsler space F n is given by (31).

Acknowledgment

The rst author is very much thankful to NBHM-DAE of government of India for their nancialassistance as a Postdoctoral Fellowship.

References

[1] C.Shibata, On invariant tensors of β -Changes of Finsler metrics, J. Math. Kyoto Univ. ,24(1984), 163-188.

[2] M.Matsumoto and S.Numata, On Finsler spaces with cubic metric, Tensor, N. S. , Vol.33(1979),153-162.

[3] M.Matsumoto, On Finsler spaces with Randers metric and special form of important ten-sors, J. Math. Kyoto Univ. , 14(3), 1974, 477-498.

[4] H.Shimada, On Finsler spaces with the metric L = {a i 1 i 2 ··· i m (x)yi 1 yi 2 · · ·yi m }1

m , Tensor,

N.S. , 33 (1979), 365-372.[5] G.Randers, On an asymmetric metric in the four-space of general Relativity, Phys. Rev. ,(2)59 (1941), 135-199.

[6] M.Matsumoto, On some transformations of locally Minkowskian space, Tensor N. S. ,Vol.22, 1971, 103-111.

[7] M.Hashiguchi and Y.Ichijyo, On some special ( α, β ) metrics, Rep. Fac. Sci. Kayoshima Univ. (Math. Phy. Chem.), Vol.8, 1975, 39-46.




[8] M.Hashiguchi and Y.Ichijyo, Randers spaces with rectilinear geodesics, Rep. Fac. Sci.Kayoshima Univ. (Math. Phy. Chem.), Vol.13, 1980, 33-40.

[9] R.S.Ingarden, On the geometrically absolute optimal representation in the electron micro-scope, Trav. Soc. Sci. Letter , Wroclaw, B45, 1957.

[10] S.Numata, On the torsion tensor Rhjk and P hjk of Finsler spaces with a metric ds ={gij (x)dx i dx j }

12 + bi (x)dx i , Tensor, N. S. , Vol.32, 1978, 27-31.

[11] P.L.Antonelli, R.S.Ingarden and M.Matsumoto, The Theory of Sparys and Finsler Spaces with Applications in Physics and Biology , Kluwer Academic Publications, Dordecht /Boston/London, 1993.

[12] I.W.Roxburgh, Post Newtonian tests of quartic metric theories of gravity, Rep. on Math.Phys. , 31(1992), 171-178.

[13] M.Matsumoto, Foundation of Finsler Geometry and Special Finsler Spaces , KaiseishaPress, Otsu, Japan, 1986.




Quarter-Symmetric Metric Connection

On Pseudosymmetric Lorentzian α −Sasakian Manifolds

C.Patra

(Govt.Degree College, Kamalpur, Dhalai, Tripura, India)

A.Bhattacharyya

(Department of Mathematics, Jadavpur University, Kolkata-700032, India)


Abstract : The object of this paper is to introduce a quarter-symmetric metric connec-

tion in a pseudosymmetric Lorentzian α-Sasakian manifold and to study of some proper-ties of it. Also we shall discuss some properties of the Weyl-pseudosymmetric Lorentzianα − Sasakian manifold and Ricci-pseudosymmetric Lorentzian α − Sasakian manifold with re-spet to quarter-symmetric metric connection. We have given an example of pseudosymmetricLorentzian α-Sasakian manifold with respect to quarter-symmetric metric connection.

Key Words : Lorentzian α -Sasakian manifold, quarter-symmetric metric connec-tion, pseudosymmetric Lorentzian α− Sasakian manifolds, Ricci-pseudosymmetric, Weyl-pseudosymmetric, η− Einstein manifold.

AMS(2010) : 53B30, 53C15, 53C25

§1. Introduction

The theory of pseudosymmetric manifold has been developed by many authors by two ways.One is the Chaki sense [8], [3] and another is Deszcz sense [2], [9], [11]. In this paper weshall study some properties of pseudosymmetric and Ricci-symmetric Lorentzian α− Sasakianmanifolds with respect to quarter-symmetric metric connection in Deszcz sense. The notionof pseudo-symmetry is a natural generalization of semi-symmetry, along the line of spaces of constant sectional curvature and locally symmetric space.

A Riemannian manifold ( M, g ) of dimension n is said to be pseudosymmetric if the Rie-mannian curvature tensor R satises the conditions ([1]):

1. (R(X, Y ).R)(U,V,W ) = LR [((X Y ).R)(U,V,W )] (1)

for all vector elds X,Y,U,V, W on M , whereLR C ∞ (M ), R(X, Y )Z = [X,Y ]Z −[ X , Y ]Z and X Y is an endomorphism dened by

(X Y )Z = g(Y, Z )X −g(X, Z )Y (2)1 Received October 6, 2012. Accepted March 6, 2013.



Quarter-Symmetric Metric Connection on Pseudosymmetric Lorentzian α − Sasakian Manifolds 47

2. (R(X, Y ).R)(U,V,W ) = R(X, Y )(R(U, V )W ) −R(R(X, Y )U, V )W

−R(U, R(X, Y )V )W −R(U, V )(R(X, Y )W ) (3)

3. ((X Y ).R)(U,V,W ) = ( X Y )(R(U, V )W ) −R((X Y )U, V )W

−R(U, (X

Y )V )W

−R(U, V )(( X

Y )W ). (4)

M is said to be pseudosymmetric of constant type if L is constant. A Riemannian manifold(M, g ) is called semi-symmetric if R.R = 0 , where R.R is the derivative of R by R.

Remark 1.1 We know, the (0 , k + 2) tensor elds R.T and Q(g, T ) are dened by

(R.T )(X 1 , · · · , X k ; X, Y ) = ( R(X, Y ).T )(X 1 , · · · , X k )

= −T (R(X, Y )X 1 , · · · , X k ) −· · ·−T (X 1 , · · · , R(X, Y )X k )

Q(g, T )(X 1 , · · · , X k ; X, Y ) = −((X Y ).T )(X 1 , · · · , X k )

= T ((X Y )X 1 , · · · , X k ) + · · ·+ T (X 1 , · · · , (X Y )X k ),

where T is a (0, k) tensor eld ([4],[5]).

Let S and r denote the Ricci tensor and the scalar curvature tensor of M respectively. Theoperator Q and the (0 , 2)−tensor S 2 are dened by

S (X, Y ) = g(QX,Y ) (5)

and

S 2(X, Y ) = S (QX,Y ) (6)

The Weyl conformal curvature operator C is dened by

C (X, Y ) = R(X, Y ) − 1n −2

[X QY + QX Y − rn −1

X Y ]. (7)

If C = 0 , n ≥ 3 then M is called conformally at. If the tensor R.C and Q(g, C ) are linearlydependent then M is called Weyl-pseudosymmetric. This is equivalent to

R.C (U,V,W ; X, Y ) = LC [((X Y ).C )(U, V )W ], (8)

holds on the set U C = {x M : C = 0 at x}, where LC is dened on U C . If R.C = 0 , then M is called Weyl-semi-symmetric. If C = 0 , then M is called conformally symmetric ([6],[10]).

§2. Preliminaries

A n-dimensional differentiable manifold M is said to be a Lorentzian α−Sasakian manifold if it admits a (1 , 1)−tensor eld φ, a contravariant vector eld ξ, a covariant vector eld η andLorentzian metric g which satisfy the following conditions,

φ2 = I + η ξ, (9)



48 C.Patra and A.Bhattacharyya

η(ξ ) = −1, φξ = 0 , η ◦φ = 0 , (10)

g(φX,φY ) = g(X, Y ) + η(X )η(Y ), (11)

g(X, ξ ) = η(X ) (12)

and( X φ)(Y ) = α{g(X, Y )ξ + η(Y )X } (13)

for X, Y χ (M ) and for smooth functions α on M, denotes covariant differentiationoperator with respect to Lorentzian metric g ([6], [7]).

For a Lorentzian α−Sasakian manifold,it can be shown that ([6],[7]):

X ξ = αφX, (14)

( X η)Y = αg(φX,Y ). (15)

Further on a Lorentzian α−Sasakian manifold, the following relations hold ([6])

η(R(X, Y )Z ) = α2[g(Y, Z )η(X ) −g(X, Z )η(Y )], (16)

R(ξ, X )Y = α2[g(Y, Z )ξ −η(Y )X ], (17)

R(X, Y )ξ = α2[η(Y )X −η(X )Y ], (18)

S (ξ, X ) = S (X, ξ ) = ( n −1)α 2 η(X ), (19)

S (ξ, ξ ) = −(n −1)α 2 , (20)

Qξ = ( n −1)α 2ξ. (21)

The above relations will be used in following sections.

§3. Quarter-Symmetric Metric Connection on Lorentzian α

−Sasakian Manifold

Let M be a Lorentzian α-Sasakian manifold with Levi-Civita connection and X , Y, Z χ (M ).We dene a linear connection D on M by

DX Y = X Y + η(Y )φ(X ) (22)

where η is 1−form and φ is a tensor eld of type (1 , 1). D is said to be quarter-symmetricconnection if T , the torsion tensor with respect to the connection D, satises

T (X, Y ) = η(Y )φX −η(X )φY. (23)

D is said to be metric connection if

(DX g)(Y, Z ) = 0 . (24)

A linear connection D is said to be quarter-symmetric metric connection if it satises (22), (23)and (24).

Now we shall show the existence of the quarter-symmetric metric connection D on aLorentzian α−Sasakian manifold M.






where S and S are the Ricci tensors of the connections D and respectively.Again

S 2(X, Y ) = S 2(X, Y ) −α(n −2)S (X, Y ) −α 2(n −1)g(X, Y )

+ α2n(n −1)(α −1)η(X )η(Y ). (33)

Contracting (32), we getr = r, (34)

where r and r are the scalar curvature with respect to the connection D and respectively.Let C be the conformal curvature tensors on Lorentzian α− Sasakian manifolds with

respect to the connections D . Then

C (X, Y )Z = R(X, Y )Z − 1n −2

[S (Y, Z )X −g(X, Z ) QY + g(Y, Z ) QX

−S (X, Z )Y ] + r

(n −1)(n −2) [g(Y, Z )X −g(X, Z )Y ], (35)

where Q is Ricci operator with the connection D on M and

S (X, Y ) = g( QX,Y ), (36)

S 2(X, Y ) = S ( QX,Y ). (37)

Now we shall prove the following theorem.

Theorem 4.1 Let M be a Lorentzian α−Sasakian manifold with respect to the quarter-symmetric metric connection D, then the following relations hold:

R(ξ, X )Y = α2 [g(X, Y )ξ −η(Y )X ] + αη(Y )[X + η(X )ξ ], (38)

η( R(X, Y )Z ) = α2 [g(Y, Z )η(X ) −g(X, Z )η(Y )], (39)

R(X, Y )ξ = ( α2 −α)[η(Y )X −η(X )Y ], (40)

S (X, ξ ) = S (ξ, X ) = ( n −1)(α 2 −α)η(X ), (41)

S 2(X, ξ ) = S 2(ξ, X ) = α2 (n −1)2(α −1)2η(X ), (42)

S (ξ, ξ ) = −(n −1)(α 2 −α), (43)

QX = QX −α(n −1)X, (44)

Qξ = ( n −1)(α 2 −α)ξ. (45)

Proof Since M is a Lorentzian α−Sasakian manifold with respect to the quarter-symmetricmetric connection D, then replacing X = ξ in (31) and using (10) and (17) we get (38). Using(10) and (16), from (31) we get (39). To prove (40), we put Z = ξ in (31) and then we use(18). Replacing Y = ξ in (32) and using (19) we get (41). Putting Y = ξ in (33) and using (6)and (19) we get (42). Again putting X = Y = ξ in (32) and using (20) we get (43). Using (36)and (41) we get (44). Then putting X = ξ in (44) we get (45).




§5. Lorentzian α−Sasakian Manifold with Respect to the Quarter-SymmetricMetric Connection D Satisfying the Condition C. S = 0 .

In this section we shall nd out the characterization of Lorentzian α−Sasakian manifold with

respect to the quarter-symmetric metric connection D satisfying the condition C. S = 0 . Wedene C. S = 0 on M by

( C (X, Y ).S )(Z, W ) = −S ( C (X, Y )Z, W ) − S (Z, C (X, Y )W ), (46)

where X,Y,Z,W χ (M ).

Theorem 5.1 Let M be an n−dimensional Lorentzian α− Sasakian manifold with respect tothe quarter-symmetric metric connection D. If C.S = 0 , then

1n −2

S 2(X, Y ) = [( α 2 −α) + r

(n −1)(n −2)]S (X, Y )

+ α2

−αn −2

[α(n −1)(α −n + 1) − r ]g(X, Y )

−α(n −1)(α 2 −α)η(X )η(Y ). (47)

Proof Let us consider M be an n-dimensional Lorentzian α−Sasakian manifold withrespect the quarter-symmetric metric connection D satisfying the condition C. S = 0 . Thenfrom (46), we get

S ( C (X, Y )Z, W ) + S (Z, C (X, Y )W ) = 0 , (48)

where X,Y,Z,W χ (M ). Now putting X = ξ in (48), we get

S ( C (ξ, X )Y, Z ) + S (Y, C (ξ, X )Z ) = 0 . (49)

Using (35), (37), (38) and (41), we have

S ( C (ξ, X )Y, Z ) = ( n −1)(α2 −α)[α 2 − (n −1)(α 2 −α)

n −2 +

r(n −1)(n −2)

]η(Z )g(X, Y )

+[ α −α 2 + (n −1)(α 2 −α)

n −2 − r

(n −1)(n −2)]η(Y )S (X, Z )

+ α(α 2 −α)(n −1)η(X )η(Y )η(Z )

− 1

n −2[(n −1)(α 2 −α)η(Z )S (X, Y ) − S 2(X, Z )η(Y )] (50)

and

S (Y, C (ξ, X )Z ) = ( n −1)(α2 −α)[α 2 − (n −1)(α 2 −α)

n −2 + r

(n −1)(n −2)]η(Y )g(X, Z )

+[ α −α 2 + (n −1)(α 2 −α)

n −2 − r

(n −1)(n −2)]η(Z )S (Y, X )

+ α(α 2 −α)(n −1)η(X )η(Y )η(Z )

− 1

n −2[(n −1)(α 2 −α)η(Y )S (X, Z ) − S 2(X, Y )η(Z )]. (51)




Using (50) and (51) in (49), we get

(n −1)(α 2 −α)[α 2 − (n −1)(α 2 −α)

n −2 +

r(n −1)(n −2)

][g(X, Y )η(Z )

+ g(X, Z )η(Y )] + 2α(α2

−α)(n

−1)η(X )η(Y )η(Z )

+[ α −α2 + (n −1)(α2 −α)

n −2 − r

(n −1)(n −2)][η(Y )S (X, Z ) + η(Z )S (Y, X )]

− 1

n −2[(n −1)(α 2 −α){η(Z )S (X, Y ) + η(Y )S (X, Z )}

−{S 2(X, Z )η(Y ) + S 2(X, Y )η(Z )}] = 0. (52)

Replacing Z = ξ in (52) and using (41) and (42), we get

1n −2

S 2(X, Y ) = [( α 2 −α) + r

(n −1)(n −2)]S (X, Y )

+α 2 −αn

−2

[α(n −1)(α −n + 1) − r ]g(X, Y )

−α(n −1)(α2 −α)η(X )η(Y ).

An n−dimensional Lorentzian α−Sasakian manifold M with the quarter-symmetric metricconnection D is said to be η−Einstein if its Ricci tensor S is of the form

S (X, Y ) = Ag(X, Y ) + Bη (X )η(Y ), (53)

where A, B are smooth functions of M. Now putting X = Y = ei , i = 1 , 2, · · · , n in (53) andtaking summation for 1 ≤ i ≤ n we get

An −B = r. (54)

Again replacing X = Y = ξ in (53) we have

A −B = ( n −1)(α 2 −α). (55)

Solving (54) and (55) we obtain

A = rn −1 −(α 2 −α) and B =

rn −1 −n(α 2 −α).

Thus the Ricci tensor of an η−Einstein manifold with the quarter-symmetric metric connectionD is given by

S (X, Y ) = [ r

n −1 −(α2 −α)]g(X, Y ) + [ r

n −1 −n(α 2 −α)]η(X )η(Y ). (56)

§6. η−Einstein Lorentzian α−Sasakian Manifold with Respect to theQuarter-Symmetric Metric Connection D Satisfying the Condition C. S = 0 .

Theorem 6.1 Let M be an η−Einstein Lorentzian α−Sasakian manifold of dimension. Then C. S = 0 iff

nα −2αnα 2 −2α

[η( R (X, Y )Z )η(W ) + η( R(X, Y )W )η(Z )] = 0 ,




where X,Y,Z,W χ (M ).

Proof Let M be an η−Einstein Lorentzian α−Sasakian manifold with respect to thequarter-symmetric metric connection D satisfying C. S = 0 . Using (56) in (48), we get

η( C (X, Y )Z )η(W ) + η( C (X, Y )W )η(Z ) = 0 ,

or,nα −2αnα 2 −2α

[η( R (X, Y )Z )η(W ) + η( R(X, Y )W )η(Z )] = 0 .

Conversely, using (56) we have

( C (X, Y ).S )(Z, W ) = −[ r

n −1 −n(α 2 −α)][η( C (X, Y )Z )η(W ) + η( C (X, Y )W )η(Z )]

= − nα −2αnα 2 −2α

[η( R (X, Y )Z )η(W ) + η( R (X, Y )W )η(Z )] = 0 .

§7. Ricci Pseudosymmetric Lorentzian α−Sasakian Manifolds withQuarter-Symmetric Metric Connection D

Theorem 7.1 A Ricci pseudosymmetric Lorentzian α- Sasakian manifolds M with quarter-symmetric metric connection D with restriction Y = W = ξ and LS = 1 is an η−Einstein manifold.

Proof Lorentzian α−Sasakian manifolds M with quarter-symmetric metric connection Dis called a Ricci pseudosymmetric Lorentzian α−Sasakian manifolds if

( R (X, Y ).S )(Z, W ) = LS [((X Y ).S )(Z, W )], (57)

or,

S ( R(X, Y )Z, W ) + S (Z, R(X, Y )W ) = LS [S ((X Y )Z, W ) + S (Z, (X Y )W )]. (58)

Putting Y = W = ξ in (58) and using (2), (38) and (41), we have

L S [S (X, Z ) −(n −1)(α2 −α)g(X, Z )]

= ( α 2 −α)S (X, Z ) −α 2(α 2 −α)(n −1)g(X, Z ) −α(α 2 −α)(n −1)η(X )η(Z ). (59)

Then for L S = 1 ,

(α 2 −α −1)S (X, Z ) = ( α2 −α)(n −1)[(α 2 −1)g(X, Z ) + αη(X )η(Z )].

Thus M is an η−Einstein manifold.

Corollary 7.1 A Ricci semisymmetric Lorentzian α-Sasakian manifold M with quarter-symmetric metric connection D with restriction Y = W = ξ is an η−Einstein manifold.




Proof Sine M is Ricci semisymmetric Lorentzian α-Sasakian manifolds with quarter-symmetric metric connection D, then L C = 0 . Putting L C = 0 in (59) we get

S (X, Z ) = α2(n −1)g(X, Z ) + α(n −1)η(X )η(Z ).

§8. Pseudosymmetric Lorentzian α−Sasakian Manifold and Weyl-pseudosymmetricLorentzian α−Sasakian Manifold with Quarter-Symmetric Metric Connection

In the present section we shall give the denition of pseudosymmetric Lorentzian α−Sasakianmanifold and Weyl-pseudosymmetric Lorentzian α−Sasakian manifold with quarter-symmetricmetric connection and discuss some properties on it.

Denition 8.1 A Lorentzian α−Sasakian manifold M with quarter-symmetric metric connec-tion D is said to be pseudosymmetric Lorentzian α−Sasakian manifold with quarter-symmetric metric connection if the curvature tensor R of M with respect to D satises the conditions

( R (X, Y ).R)(U,V,W ) = LR [((X Y ).R )(U,V,W )], (60)

where

( R(X, Y ).R )(U,V,W ) = R (X, Y )( R(U, V )W ) − R ( R (X, Y )U, V )W

−R(U, R(X, Y )V )W − R (U, V )(R(X, Y )W ), (61)

and

((X Y ).R )(U,V,W ) = ( X Y )( R (U, V )W ) − R((X Y )U, V )W

−R(U, (X

Y )V )W

− R(U, V )(( X

Y )W ). (62)

Denition 8.2 A Lorentzian α−Sasakian manifold M with quarter-symmetric metric con-nection D is said to be Weyl- pseudosymmetric Lorentzian α−Sasakian manifold with quarter-symmetric metric connection if the curvature tensor R of M with respect to D satises the conditions

( R(X, Y ).C )(U,V,W ) = LC [((X Y ).C )(U,V,W )], (63)

where

( R(X, Y ).C )(U,V,W ) = R(X, Y )( C (U, V )W ) − C ( R(X, Y )U, V )W

−C (U, R(X, Y )V )W − C (U, V )(R(X, Y )W ) (64)

and

((X Y ).C )(U,V,W ) = ( X Y )( C (U, V )W ) − C ((X Y )U, V )W

−C (U, (X Y )V )W − C (U, V )(( X Y )W ). (65)

Theorem 8.1 Let M be an n dimensional Lorentzian α−Sasakian manifold. If M is Weyl-pseudosymmetric then M is either conformally at and M is η−Einstein manifold or LC = α2 .




Proof Let M be an Weyl-pseudosymmetric Lorentzian α−Sasakian manifold and X,Y,U,V,W χ (M ). Then using (64) and (65) in (63), we have

R(X, Y )( C (U, V )W ) − C ( R (X, Y )U, V )W

−¯C (U,

¯R(X, Y )V )W −

¯C (U, V )(R(X, Y )W )

= LC [(X Y )( C (U, V )W ) − C ((X Y )U, V )W (66)

−C (U, (X Y )V )W − C (U, V )(( X Y )W )]. (67)

Replacing X with ξ in (66) we obtain

R(ξ, Y )( C (U, V )W ) − C ( R (ξ, Y )U, V )W

−C (U, R(ξ, Y )V )W − C (U, V )(R(ξ, Y )W )

= LC [(ξ Y )( C (U, V )W ) − C ((ξ Y )U, V )W

−C (U, (ξ Y )V )W − C (U, V )(( ξ Y )W )]. (68)

Using (2), (38) in (67) and taking inner product of (67) with ξ , we get

α 2 [−C (U,V,W,Y ) −η( C (U, V )W )η(Y ) −g(Y, U )η( C (ξ, V )W )

+ η(U )η( C (Y, V )W ) −g(Y, V )η( C (U, ξ )W ) + η(V )η( C (U, Y )W )

+ η(W )η( C (U, V )Y )] −α[η(U )η( C (φ2Y, V )W )

+ η(V )η( C (U, φ2Y )W ) + η(W )η( C (U, V )φ2 Y )]

= LC [−C (Y,U,V,W ) −η(Y )η( C (U, V )W ) −g(Y, U )η( C (ξ, V )W )

+ η(U )η( C (Y, V )W ) −g(Y, V )η( C (U, ξ )W ) + η(V )η( C (U, Y )W )

+ η(W )η( C (U, V )Y )]. (69)

Putting Y = U, we get

[L C −α2 ][g(U, U )η( C (ξ, V )W ) + g(U, V )η( C (U, ξ )W )] + αη(V )η( C (φ2 U, V )W ) = 0 . (70)

Replacing U = ξ in (68), we obtain

[L C −α 2 ]η( C (ξ, V )W ) = 0 . (71)

The formula (69) gives either η( C (ξ, V )W ) = 0 or L C −α 2 = 0 .Now L C −α 2 = 0 , then η( C (ξ, V )W ) = 0 , then we have M is conformally at and which

givesS (V, W ) = Ag(V, W ) + Bη (V )η(W ),

whereA = [α 2 −

(n −1)(α 2 −α)n −2

+ r

(n −1)(n −2)](n −2)

andB = [α 2 −

2(n −1)(α 2 −α)n −2

+ r

(n −1)(n −2)](n −2),

which shows that M is an η−Einstein manifold. Now if η( C (ξ, V )W ) = 0 , then L C = α2 .






where α is a constant.Clearly, {e1 , e2 , e3} is a set of linearly independent vectors for each point of M and hence

a basis of χ (M ). The Lorentzian metric g is dened by

g(e1 , e2) = g(e2 , e3) = g(e1 , e3) = 0 ,g(e1 , e1) = g(e2 , e2) = g(e3 , e3) = −1.

Then the form of metric becomes

g = 1(ex 3 )2 (dx2)2 −

1α 2 (dx3 )2 ,

which is a Lorentzian metric.Let η be the 1−form dened by η(Z ) = g(Z, e 3) for any Z χ (M ) and the (1 , 1)−tensor

eld φ is dened byφe1 = −e1 , φe2 = −e2 , φe3 = 0 .

From the linearity of φ and g, we have

η(e3) = −1,

φ2 (X ) = X + η(X )e3 and

g(φX,φY ) = g(X, Y ) + η(X )η(Y )

for any X χ (M ). Then for e3 = ξ, the structure ( φ,ξ,η,g) denes a Lorentzian paracontactstructure on M .

Let be the Levi-Civita connection with respect to the Lorentzian metric g. Then we have

[e1 , e2] = 0, [e1 , e3] = −αe 1 , [e2 , e3] = −αe 2 .

Koszul’s formula is dened by

2g( X Y, Z ) = Xg(Y, Z ) + Y g(Z, X ) −Zg (X, Y )

−g(X, [Y, Z ]) −g(Y, [X, Z ]) + g(Z, [X, Y ]).

Then from above formula we can calculate the followings,

e1 e1 = −αe 3 , e1 e2 = 0 , e1 e3 = −αe 1 ,

e2 e1 = 0 , e2 e2 = −αe 3 , e2 e3 = −αe 2 ,

e3 e1 = 0 , e3 e2 = 0 , e3 e3 = 0 .

Hence the structure ( φ,ξ,η,g) is a Lorentzian α−Sasakian manifold [7].Using (22), we nd D, the quarter-symmetric metric connection on M following:

D e1 e1 = −αe 3 , De1 e2 = 0 , De1 e3 = e1(1 −α),

D e2 e1 = 0 , De2 e2 = −αe 3 , De2 e3 = e2(1 −α),

D e3 e1 = 0 , De3 e2 = 0 , De3 e3 = 0 .




Using (23), the torson tensor T , with respect to quarter-symmetric metric connection D asfollows:

T (ei , ei ) = 0 , i = 1 , 2, 3

T (e1 , e2) = 0 , T (e1 , e3) = e1 , T (e2 , e3) = e2 .

Also (D e1 g)(e2 , e3) = ( D e2 g)(e3 , e1) = ( D e3 g)(e1 , e2) = 0. Thus M is Lorentzian α−Sasakianmanifold with quarter-symmetric metric connection D .

Now we calculate curvature tensor R and Ricci tensors S as follows:

R(e1 , e2)e3 = 0 , R(e1 , e3)e3 = −(α 2 −α)e1 ,

R(e3 , e2)e2 = α2 e3 , R (e3 , e1)e1 = α2 e3 ,

R(e2 , e1)e1 = ( α2 −α)e2 , R (e2 , e3)e3 = −α 2e2 ,

R(e1 , e2)e2 = ( α2 −α)e1 .

S (e1 , e1) = S (e2 , e2) =

−α and S (e3 , e3) =

−2α 2 + ( n

−1)α.

Again using (2), we get

(e1 , e2)e3 = 0 , (ei ei )ej = 0 , i, j = 1 , 2, 3,

(e1 e2)e2 = ( e1 e3)e3 = −e1 , (e2 e1)e1 = ( e2 e3)e3 = −e2,

(e3 e2)e2 = ( e3 e1)e1 = −e3 .

Now,

R(e1 , e2)( R (e3 , e1)e2) = 0 , R( R (e1 , e2)e3 , e1)e2 = 0 ,

R(e3 , R(e1 , e2)e1)e2 =

−α 2(α 2

−α)e3 ,

( R(e3 , e1)( R (e1 , e2)e2) = α2 (α 2 −α)e3 .

Therefore, ( R (e1 , e2).R)(e3 , e1 , e2) = 0.Again,

(e1 e2)( R (e3 , e1)e2) = 0 , R((e1 e2)e3 , e1)e2 = 0 ,

R(e3 , (e1 e2)e1)e2 = α2e3 , R(e3 , e1)(( e1 e2)e2) = −α 2e3 .

Then (( e1 , e2).R )(e3 , e1 , e2) = 0. Thus ( R (e1 , e2).R)(e3 , e1 , e2) = LR [((e1 , e2).R)(e3 , e1 , e2)] forany function = LR C ∞ (M ).

Similarly, any combination of e1 , e2 and e3 we can show (60). Hence M is a pseudosym-metric Lorentzian α−Sasakian manifold with quarter-symmetric metric connection.

References

[1] Samir Abou Akl, On partially pseudosymmetric contact metric three-manifold, Damascus University Journal for Basic Science , Vol.24, No 2(2008), pp 17-28.




[2] F.Defever, R.Deszcz and L.Verstraelen, On pseudosymmetric para-Kahler manifolds, Col-loquium Mathematicum , Vol. 74, No. 2(1997), pp. 153-160.

[3] K.Arslan, C.Murathan, C.Ozgur and A.Yildizi, Pseudosymmetric contact metric manifoldsin the sense of M.C. Chaki, Proc. Estonian Acad. Sci. Phys. Math. , 2001, 50, 3, 124-132.

[4] R.Deszcz, On pseudo-symmetric spaces, Bull. Soc. Math. Belg. , Ser. A, 44(1992), 1-34.[5] M.Belkhelfa, R.Deszcz and L.Verstraelen, Symmetric properties of Sasakian space forms,

Soochow Journal of Mathematics , Vol 31, No 4, pp.611-616, Oct., 2005.[6] D.G.Prakashs, C.S.Bagewadi and N.S.Basavarajappa, On pseudosymmetric Lorentzian α-

Sasakian manifolds, IJPAM , Vol. 48, No.1, 2008, 57-65.[7] A.Yildiz, M.Turan and B.F.Acet, On three dimensional Lorentzian α−Sasakian manifolds,

Bulletin of Mathematical Analysis and Applications , Vol. 1, Issue 3(2009), pp. 90-98.[8] M.C.Chaki and B.Chaki, On pseudo symmetric manifolds admitting a type of semisym-

metric connection, Soochow Journal of Mathematics , vol. 13, No. 1, June 1987, 1-7.[9] R.Deszcz, F.Defever, L.Verstraelen and L.Vraneken, On pseudosymmetric spacetimes, J.

Math. Phys. , 35(1994), 5908-5921.[10] C.Ozgur, On Kenmotsu manifolds satisfying certain pseudosymmetric conditions, World

Applied Science Journal , 1 (2); 144-149, 2006.[11] R.Deszcz, On Ricci-pseudosymmetric warped products, Demonstratio Math. , 22(1989),

1053-1065.




The Skew Energy of

Cayley Digraphs of Cyclic Groups and Dihedral Groups

C.Adiga 1 , S.N.Fathima 2 and Haidar Ariamanesh 1

1. Department of Studies in Mathematics, University of Mysore, Mysore-570006, India

2. Department of Mathematics, Ramanujan School of Mathematical Sciences,

Pondicherry University, Puducherry-605014 India

E-mail: c [email protected], [email protected], [email protected]

Abstract : This paper is motivated by the skew energy of a digraph as vitiated by C.Adiga,R.Balakrishnan and Wasin So [1]. We introduce and investigate the skew energy of a Cayleydigraphs of cyclic groups and dihedral groups and establish sharp upper bound for the same.

Key Words : Cayley graph, dihedral graph, skew energy.

AMS(2010) : 05C25, 05E18

§1. Introduction

Let G be a non trivial nite group and S be an non-empty subset of G such that for x S, x− 1 /S and I G / S , then the Cayley digraph Γ = Cay(G, S ) of G with respect to S is dened as asimple directed graph with vertex set G and arc set E (Γ) = {(g, h)|hg− 1 S }. If S is inverse

closed and doesn’t contain identity then Cay(G, S ) is viewed as undirected graph and is simplythe Cayley graph of G with respect to S . It easily follows that valency of Cay(G, S ) is |S | andCay (G, S ) is connected if and only if S = G. For an elaborate literature on Cayley graphsone may refer [5]. A dihedral group D 2n is a group with 2 n elements such that it contains anelement ′a ′ of order 2 and an element ′b′ of order n with a− 1ba = b− 1 . Thus D 2n = a, b|a2 =bn = 1 , a − 1ba = b− 1 = a, b|a2 = bn = 1 , a − 1ba = bα , α ≡ 1(mod n ), α 2 ≡ 1(mod n ) .

If n = 2, then D4 is Abelian; for n ≥ 3, D2n is not abelian. The elements of diheral groupcan be explicitly listed as

D2n = {1,a,ab,ab 2, · · · , abn − 1 , b ,b2 , · · · , bn − 1}.

In short, its elements can be listed as a i bk where i = 0 , 1 and k = 0 , 1,

· · · , (n

−1). It is easy to

explicitly describe the product of any two elements a i bk a j bl = ar bs as follows:

1. If j = 0 then r = i and s equals the remainder of k + l modulo n.

2. If j = 1 , then r is the remainder of i + j modulo 2 and s is the remainder of kα + lmodulo n.

1 Received October 16, 2012. Accepted March 8, 2013.



The Skew Energy of Cayley Digraphs of Cyclic Groups and Dihedral Groups 61

The orders of the elements in the Dihedral group D2n are: o(1) = 1, o(abi ) = 2 , where;0 ≤ i ≤ n −1, o(bi ) = n, where; 0 < i ≤ n −1 and if n is even than o(b

n2 ) = 2 .

Let Γ be a digraph of order n with vertex set V (Γ) = {v1 , · · · , vn },and arc set Λ(Γ) V (Γ) ×V (Γ). We assume that Γ does not have loops and multiple arcs, i.e.,( vi , vi ) / Λ(Γ) for

all i , and (vi , vj ) Λ(Γ) implies that ( vj , vi ) / Λ(Γ). Hence the underlying undirected graphGΓ of Γ is a simple graph. The skew-adjacency matrix of Γ is the n ×n matrix S (Γ) = [ s ij ],where sij = 1 whenever ( vi , vj ) Λ(Γ), s ij = −1 whenever ( vj , vi ) Λ(Γ), and sij = 0otherwise. Because of the assumptions on Γ, S (Γ) is indeed a skew-symmetric matrix. Hencethe eigenvalues {λ1 , · · · , λn } of S (Γ) are all purely imaginary numbers, and the singular valuesof S (Γ) coincide with the absolute values {|λ1|, · · · , |λn |} , of its eigenvalues. Consequently, theenergy of S (Γ) , which is dened as the sum of its singular values [6], is also the sum of theabsolute values of its eigenvalues. For the sake of convenience, we simply refer the energy of S (Γ) as the skew energy of the digraph Γ. If we denote the skew energy of Γ by εs (Γ) then,

εs (Γ) =n

i =1|λ i |.

The degree of a vertex in a digraph Γ is the degree of the corresponding vertex of theunderlying graph of Γ. Let D(Γ) = diag (d1 , d2 , · · · , dn ), the diagonal matrix with vertex degreesd1 , d2 , · · · , dn of v1 , v2 , · · · , vn and S (Γ) be the skew adjacency matrix of a simple digraph Γ,possessing n vertices and m edges. Then L(Γ) = D(Γ) −S (Γ) is called the Laplacian matrix of the digraph Γ. If λ i , i = 1 , 2, · · · , n are the eigenvalues of the Laplacian matrix L(Γ) then the

skew Laplacian energy of the digraph Γ is dened as SLE (Γ) =n

i =1|λ i −

2mn |.

An n ×n matrix S is said to be a circulant matrix if its entries satisfy sij = s1,j − i +1 , wherethe subscripts are reduced modulo n and lie in the set {1, 2,...,n }. In other words, ith row of S is obtained from the rst row of S by a cyclic shift of i −1 steps, and so any circulant matrix is

determined by its rst row. It is easy to see that the eigenvalues of S are λk =

n

j =1s1j ω

( j − 1) k,

k = 0 , 1, · · · , n −1. For any positive integer n, let τ n = {ωk : 0 ≤ k < n } be the set of all nthroots of unity, where ω = e

2 πin = cos( 2π

n ) + i sin( 2πn ) that i2 = −1. τ n is an abelian group with

respect to multiplication. A circulant graph is a graph Γ whose adjacency matrix A(Γ) is acirculant matrix. More details about circulant graphs can be found in [3].

Ever since the concept of the energy of simple undirected graphs was introduced by Gutmanin [7], there has been a constant stream of papers devoted to this topic. In [1], Adiga, et al. havestudied the skew energy of digraphs. In [4], Gui-Xian Tian, gave the skew energy of orientationsof hypercubes. In this paper we introduce and investigate the skew energy of a Cayley digraphsof cyclic groups and dihedral groups and establish sharp upper bound for the same.

§2. Main Results

First we present some facts that are needed to prove our main results.

Lemma 2.1([2]) Let Γ is disconnected graph into the λ components Γ1 , Γ2 , · · · , Γλ , then






Proof The proof of ( i) follows directly from Lemma 2.3( iv), and ( ii ) is a consequence of Lemma 2.3( v).

Lemma 2.5 Let n be a positive integer. Then

(i)

n − 12

k=0

sin 2kπ

n =

n − 12

k=0|sin

4kπn |, n ≡ 1(mod2),

(ii )

n − 22

k=0

sin 2kπ

n =

n − 22

k=0|sin

4kπn |, n ≡ 2(mod4),

(iii )

n − 22

k=1

sin 2kπ

n = 1 + 2

n − 44

k=1

sin 2kπ

n , n ≡ 0(mod4),

(iv)

n − 22

k=1|sin

4kπn | = 2

n − 44

k=1|sin

4kπn |, n ≡ 0(mod4),

(v)

n − 44

k=1

sin 4kπ

n = csc

4πn

+ cot 4πn

, n ≡ 0(mod8),

(vi)

n − 44

k=1

sin 4kπ

n = cot

2πn

, n ≡ 4(mod8).

Proof (i) Let n ≡ 1(mod2), f (k) = sin 2kπ

n and g(k) = sin

4kπn

, where k {0, 1, 2, · · · , n −1

2 }.Then it is easy to check that

g(k) =f (2k) if 0 ≤ k ≤ n − 1

4 ,

−f (n −2k) if n − 14 < k ≤ n − 1

2 .

This implies ( i).

(ii ) Let n

≡ 2(mod4), f (k) = sin

2kπ

n and g(k) = sin

4kπ

n , where k

{0, 1, 2,

· · · ,

n −2

2 }.

Then it follows that

g(k) =f (2k) if 1 ≤ k ≤ n − 2

4 ,

−f (−n2 + 2k) if n − 2

4 < k ≤ n − 22 .

This implies ( ii ). Proofs of ( iii ) and ( iv) are similar to that of ( i) and ( ii ).



64 C.Adiga, S.N.Fathima and Haidar Ariamanesh

(v) Let n ≡ 0(mod8). Then n = 8m, m N andn − 4

4

k=1

sin 4kπ

n =

2m − 1

k=1

sin kπ2m

= sin π2m

+ sin 3π2m

+ · · ·+ sin (2m −1)π

2m

+ sin 2π2m

+ sin 4π2m

+ · · ·+ sin (2m −2)π2m

= csc π2m

+ cot π2m

= csc 4πn

+ cot 4πn

(Using Lemma2 .3(vi) and ( ii )) .

(vi) Let n ≡ 4(mod8). Then n = 8m + 4, m N andn − 4

4

k=1

sin 4kπ

n =

2m

k=1

sin kπ2m + 1

= cot π

2(2m + 1) = cot

2π

n . (using Lemma 2 .3(vii ))


(i)n − 1

k=1|sin

2kπn | = cot

π2n

, n ≡ 1(mod2),

(ii )n − 1

k=1|sin

2kπn | = 2cot

πn

, n ≡ 0(mod2).

Now we compute skew energy of some Cayley digraphs.

Theorem 2.7 Let G = {v1 = e, v2 , · · · , vn } be a group, S = {vi} G with vi = v− 1i , vi = e

and Γ = Cay(G, S ) be a Cayley digraph on G with respect to S. Suppose H = S , |H | = m,

|G : H | = λ. Then

εs (Γ) =2λcot π

2m if m ≡ 1(mod2),

4λcot πm if m ≡ 0(mod2).

Proof Let G = {v1 = e, v2 , v3 , · · · , vn }, S = {vi}, vi G with vi = v− 1i , vi = e and suppose

H = S , |H | = m, |G : H | = λ. If λ = 1 then G = {e, vi , v2i , · · · , vn − 1

i } and hence the theskew-adjacency matrix of Γ = Cay(G, S ) is a circulant matrix. Its rst row is [0 , 1, 0, · · · , 0, −1].So all eigenvalues of Γ are λk = ωk −ωkn − k = ωk −ω− k = 2 isin 2kπ

n , k = 0 , 1, · · · , n −1 whereω = e

2 πin and i2 =

−1. Applying Lemma 2.2( ii ), we obtain λk = 2 isin 2kπ

n , k = 0 , 1,

· · · , n

−1.

Now by Lemma 2.6 we have

εs (Γ) =n − 1

k=0|2isin

2kπn | = 2

n − 1

k=1|sin

2kπn | =

2cot π2n if n ≡ 1(mod2),

4cot πn if n ≡ 0(mod2).

If λ > 1, then Γ is disconnected graph in to the Γ i , i = 1 , · · · , λ components and allcomponents are isomorphic with Cayley digraph Γ m = C ay(H, S ) where H = vi : vm

i = 1




and m|n, S = {vi}. Since Γ is not connected, by Lemma 2.1, its energy is the sum of theenergies of its connected components. Thus

εs (Γ) =λ

i =1

εs (Γ i ) = λε s (Cay (H, S )) =2λcot π

2m if m ≡ 1(mod2),

4λcot πm if m ≡ 0(mod2).

This completes the proof.

Theorem 2.8 Let G = {e,b,b2, · · · , bn − 1} be a cyclic group of order n , and Γ = Cay(G, S ) be a Cayley digraph on G with respect to S = {bi , bj }, 0 < i, j ≤ n −1, i = j H = S , |H | = m,and |G : H | = λ. Then

(i) εs (Γ) ≤ 4λ cot π2m if m ≡ 1(mod2),

(ii ) εs (Γ) ≤ 8λ cot πm if m ≡ 2(mod4),

(iii ) εs(Γ)

≤ 4λ(cot π

m + 2 csc 4π

m + 2cot 4π

m) if m

≡ 0(mod8),

(iv) εs (Γ) ≤ 4λ(cot πm + 2 cot 2π

m )) if m ≡ 4(mod8).

Proof Let G = {e,b,b2, · · · , bn − 1} be a cyclic group of order n and Γ = C ay(G, S ) be aCayley digraph on G with respect to S = {bi , bj }, 0 < i, j ≤ n −1, i = j, H = S , |H | = m, and

|G : H | = λ. If λ = 1 , then G = H and hence the the skew-adjacency matrix of Γ = Cay(G, S )is a circulant matrix. So all eigenvalues of Γ are λk = ωk −ω− k + ω2k −ω− 2k , k = 0 , 1, · · · , n −1,where ω = e

2 πin and i2 = −1. Hence

λk = ωk −ω− k + ω2k −ω− 2k = 2 isin2kπ

n + 2 isin

4kπn

= 2 i(sin2kπ

n + sin

4kπn

)

for k = 0 , 1, · · · , n −1.

(i) Suppose n ≡ 1(mod2). Then

εs (Γ) =n − 1

k=0|λk | =

n − 1

k=0|2i(sin

2kπn

+ sin4kπ

n )|

=n − 1

k=1|2i(sin

2kπn

+ sin4kπ

n )|

= 4

n − 12

k=1|sin

2kπn

+ sin4kπ

n |

≤ 4(n − 12

k=1|sin

2kπn |+

n − 12

k=1|sin

4kπn |)

= 4(

n − 12

k=1

sin2kπ

n +

n − 12

k=1

sin2kπ

n ) (using Lemma 2 .5(i))

= 4 cot π2n

(applying Lemma 2 .3(i)) .




Thus ( i) holds for λ = 1.

Suppose λ > 1. Then Γ is disconnected graph in to the Γ i , i = 1 , · · · , λ components and allcomponents are isomorphic with Cayley digraph Γ m = Cay(H, S ) where H = vi : vm

i = 1 andm|n, S = {vi}. Since Γ is not connected, its energy is the sum of the energies of its connected

components. Thus

εs (Γ) =λ

i=1

εs (Γ i )λε s (Cay (H, S )) ≤ 4λ cot π2m

.

Now we shall prove ( ii ),( iii ) and ( iv) only for λ = 1 . For λ > 1, proofs are similar to thatof (i).

(ii ) If n ≡ 2(mod4), then

εs (Γ) =n − 1

k=0

|λk | =n − 1

k=0

|2i(sin2kπ

n + sin

4kπn

)|

=n − 1

k=1|2i(sin

2kπn

+ sin4kπ

n )|

= 4

n − 22

k=1|sin

2kπn

+ sin4kπ

n |

≤ 4(

n − 22

k=1|sin

2kπn |+

n − 22

k=1|sin

4kπn |)

= 4(

n − 22

k=1

sin2kπ

n +

n − 22

k=1

sin2kπ

n ) (using Lemma 2 .5(ii ))

= 8 cot πn

.

Here we used the Lemma 2 .3(ii ).

(iii ) If n ≡ 0(mod8), then

εs (Γ) = 4

n − 22

k=1|sin

2kπn

+ sin4kπ

n |

≤ 4(

n − 22

k=1|sin

2kπn |+

n − 22

k=1|sin

4kπn |)

= 4(

n − 22

k=1

sin2kπ

n + 2

n − 44

k=1

sin4kπ

n ) (using Lemma 2 .5(iv))

= 4 cotπn

+ 8 csc4πn

+ 8cot4πn

.

To get the last equality we have used Lemma 2 .3(ii ) and 2 .5(v).




(iv) If n ≡ 4(mod8), then

εs (Γ) ≤ 4(

n − 22

k=1|sin

2kπn |+

n − 22

k=1|sin

4kπn |)

= 4(n − 2

2

k=1

sin2kπ

n + 2

n − 44

k=1

sin4kπ

n ) (using Lemma 2 .5(iv))

= 4( cotπn

+ 2 cot2πn

)).

To get the last equality we have used Lemma 2 .3(ii ) and 2 .5(vi).

Lemma 2.9 Let G = b : bn = 1 be a cyclic group and Γ = Cay(G, S t ), t {1, · · · , n − 12 },be

a Cayley digraph on G with respect to S t = {bl , b2l , · · · , btl }, where l U (n) = {r : 1 ≤ r <n,gcd(n, r ) = 1}. Then the eigenvalues of Γ are

λk =| S t |

j =0

2isin 2kjπn

, k = 0 , 1, · · · , n −1,

where i2 = −1.

Proof The proof directly follows from the denition of cyclic group and is similar to thatof Theorem 2 .7.

Lemma 2.10 Let G = b : bn = 1 be a cyclic group and Γ = Cay(G, S t ), t {1, · · · , n − 12 },be

a Cayley digraph on G with respect to S t = {bl , b2l , · · · , btl } where l U (n) = {r : 1 ≤ r <n,gcd(n, r ) = 1}. Also suppose εs (Γ) ,SLE (Γ) denote the skew energy and the skew Laplacian

energy of Γ respectively. Then εs (Γ) = SLE (Γ) .Proof The proof directly follows from the denition of the skew energy and the skew

Laplacian energy.


(i)

n − 12

k=1

cos 4kπ

n =

n − 12

k=1

cos 2kπ

n , n ≡ 1(mod2),

(ii )

n − 22

k=1 |cos

4kπ

n | = csc

π

n −1, n

≡ 2(mod4),

(iii )

n − 22

k=1

cos 4kπ

n = −1, n ≡ 0(mod4).

Proof (i) Let n ≡ 1(mod2), f (k) = cos 2kπ

n , g(k) = cos

4kπn

, where k {1, 2, · · · , n −1

2 }.




It is easy to verify that

g(k) =f (2k) if 1 ≤ k ≤ n − 1

4 ,

f (n −2k) if n − 14 < k ≤ n − 1

2 .

This implies ( i).

(ii ) Let n ≡ 2(mod4). Then n = 4m + 2 for some m N . We have

n − 22

k=1|cos

4kπn | =

n − 22

k=1|cos

4kπn | =

2m

k=1|cos

4kπ4m + 2 |

=2m

k=1|cos

2kπ2m + 1 | = csc

π2(2m + 1) −1 (using Lemma 2 .3(viii )

= csc πn −1.

(iii ) Suppose n = 4m, m N . Then

n − 22

k=1

cos 4kπ

n =

2m − 1

k=1

coskπm

= cos mπ

m +

m − 1

k=1

coskπm

+2m − 1

k= m +1

coskπm

.

Changing k to k + m in the last summation we get

n − 22

k=1

cos 4kπ

n = −1.

Theorem 2.12 Let G = b : bn = 1 be a cyclic group and Γ = Cay(G, S ),be a Cayley digraph on G with respect to S = {bl} where l U (n) = {r : 1 ≤ r < n,gcd (n, r ) = 1} and C s (Γ) be

the skew-adjacency matrix of Γ , D(Γ) = diag (d1 , d2 , · · · , dn ), the diagonal matrix with vertex degrees d1 , d2 , · · · , dn of e,b,b2 , · · · , bn − 1 . Suppose L(Γ) = D(Γ) − C s (Γ) and µ1 , · · · , µn are

eigenvalues of L(Γ) . We dene α(Γ) =n

i =1

µ2i . Then

(i) α(Γ) ≤ 2n + 2 csc π2n if n ≡ 1(mod2),

(ii ) α(Γ) ≤ 2(n −1) + 4 csc πn if n ≡ 2(mod4),

(iii ) α(Γ) = 2( n −2) if n ≡ 4(mod0).

Proof Let G = b : bn

= 1 be a cyclic group and Γ = Cay (G, S ),be a Cayley digraphon G with respect to S = {bl} where l U (n) = {r : 1 ≤ r < n,gcd (n, r ) = 1} and C s (Γ)be the skew-adjacency matrix of Γ . Note that underlying graph of Γ is a 2 −regular graph.Hence D (Γ) = diag (2, 2, · · · , 2). Suppose L(Γ) = D(Γ) −C s (Γ) then L(Γ) is a circulant matrixand its rst row is [2 , −1, 0, · · · , 0, 1]. This implies that the eigenvalues of L(Γ) are µk =2 −ωk + ωkn − k = 2 −ωk + ω− k = 2 −(ωk −ω− k ) = 2 −2i sin 2kπ

n , k = 0 , 1, · · · , n −1 whereω = e

2 πin and i2 = −1.






If n ≡ 2(mod4), then

α(Γ) ≤ 4 + 4(n −2

2 ) + 4

n − 22

k=1|cos

4kπn |

= 4 + 4( n −12

) + 4(csc πn −1) (using Lemma2 .11(ii ))

= 2( n −1) + 4 csc πn

.

This completes the proof of ( ii ).If n ≡ 0(mod4), then

α(Γ) = 4 + 4(n −2

2 ) + 4

n − 22

k=1

cos 4kπ

n

= 4 + 4(n −2

2 ) + 4(−1) (using Lemma2 .11(iii ))

= 2( n −2).This completes the proof of ( iii ).


(i)

n − 12

k=1|cos

6kπn | =

32

csc 3π2n

+ 12

, n ≡ 3(mod6),

(ii )

n − 12

k=1|cos

6kπn | =

n − 12

k=1|cos

2kπn |, n ≡ 1(mod6),

(iii )

n − 12

k=1|cos

6kπn | =

n − 12

k=1|cos

2kπn |, n ≡ 5(mod6),

(iv)

n − 22

k=1

cos 6kπ

n = 0, n ≡ 0(mod6),

(v)

n − 22

k=1

cos 6kπ

n =

n − 22

k=1

cos 2kπ

n , n ≡ 2(mod6),

(vi)

n − 22

k=1

cos 6kπ

n

=

n − 22

k=1

cos 2kπ

n

, n

≡ 4(mod6),

vii)

n − 12

k=1

cos 8kπ

n =

n − 12

k=1

cos 4kπ

n , n ≡ 1(mod2),

(vii )

n − 22

k=1

cos 8kπ

n =

n − 22

k=1

cos 4kπ

n , n ≡ 2(mod4),




(viii )

n − 22

k=1|cos

8kπn | = −1 + 2csc

2πn

, n ≡ 4(mod8),

(ix )

n − 22

k=1|cos 8kπ

n | = −1 + 4cot 4πn

, n ≡ 0(mod16),

(x)

n − 22

k=1|cos

8kπn | = −1 + 4csc

4πn

, n ≡ 8(mod16) and n ≡ 2 or 4(mod6).

Proof (i) Suppose n = 6m + 3 , then

n − 12

k=1 |cos

6kπ

n | =

3m +1

k=1 |cos

2kπ

2m + 1 |=

2m

k=1|cos

2kπ2m + 1 |+

3m +1

k=2 m +1|cos

2kπ2m + 1 |

=2m

k=1|cos

2kπ2m + 1 |+

m

k=0|cos

2kπ2m + 1 |

(changing k to k + (2m + 1) in the last summation)

= 3

2 csc

3π2n

+ 12

(using Lemma 2 .3(viii ), (iii )) .

(ii ) Let n = 6m + 1, f (k) = cos 2kπ

n

, g(k) = cos 6kπ

n

, where k

{1, 2,

· · · ,

n −1

2 }. Then

we have

g(k) =

f (3k) if 1 ≤ k ≤ m,

f (n −3k) if m < k ≤ 2m,

f (3k −n) if 2m < k ≤ 3m.

This implies ( ii ).

The proofs of ( iii ), (iv), (v), (vi), (vii ) and ( viii ) are similar to the proof of ( ii ).

Suppose n = 4m then

n − 22

k=1|cos 8kπn | =

2m − 1

k=1|cos 2kπm |

= 1 +m − 1

k=1|cos

2kπm |+

2m − 1

k= m +1|cos

2kπm |

= 1 + 2m − 1

k=1|cos

2kπm | = −1 + 2

m − 1

k=0|cos

2kπm |




=−1 + 2 csc π

2m if m ≡ 1(mod2)

−1 + 4 cot πm if m ≡ 0(mod4)

−1 + 4 csc πm if m ≡ 2(mod4)

= −1 + 2 csc 2π

n if n ≡ 4(mod8),

−1 + 4 cot 4πn if n ≡ 0(mod16),

−1 + 4 csc 4πn if n ≡ 8(mod16).

This completes the proof of ( viii ),( ix ),(x).

Theorem 2.14 Let G = b : bn = 1 be a cyclic group and Γ = Cay(G, S ),be a Cayley digraph on G with respect to S = {bl , b2l} where l U (n) = {r : 1 ≤ r < n, gcd (n, r ) = 1} and C s (Γ) be the skew-adjacency matrix of Γ , D(Γ) = diag (d1 , d2 , · · · , dn ), the diagonal matrix with vertex degrees d1 , d2 , · · · , dn of e,b,b2 , · · · , bn − 1 . Suppose L(Γ) = D(Γ) − C s (Γ) and µ1 , · · · , µn are

eigenvalues of L(Γ) . Dene α (Γ) =n

i=1

µ2i . Then

(i) α(Γ) ≤ 4(3n + 2) −12csc 3π2n if n ≡ 3(mod6).

(ii ) α(Γ) ≤ 4(3n + 2) −4csc π2n if n ≡ 1 or 5(mod6).

(iii ) α(Γ) ≤ 4(3n −2) + 16csc πn if n ≡ 2(mod4) and n ≡ 0(mod6).

(iv) α(Γ) ≤ 4(3n −2) + 24csc πn if n ≡ 2(mod4)and n ≡ 2 or 4(mod6).

(v) α(Γ) ≤ 4(3n −2) + 8cot πn + 8 csc 2π

n if n ≡ 4(mod8) and n ≡ 0(mod6).

(vi) α(Γ) ≤ 4(3n −4) + 16cot πn + 8 csc 2π

n if n ≡ 4(mod8) and n ≡ 2 or 4(mod6).

(vii ) α(Γ) ≤ 4(3n −2) + 8cot πn + 16cot 4πn if n ≡ 0(mod16), and n ≡ 0(mod6).

(viii ) α(Γ) ≤ 2(n −8) + 16cot πn + 16cot 4π


(ix ) α(Γ) ≤ 4(3n −2) + 8cot πn + 16csc 4π


(x) α(Γ) ≤ 4(3n −4) + 16cot πn + 16 csc 4π


Proof Let G = b : bn = 1 be a cyclic group and Γ = Cay(G, S ),be a Cayley digraph onG with respect to S = {bl , b2l} where l U (n) = {r : 1 ≤ r < n, gcd (n, r ) = 1} and C s (Γ) bethe skew-adjacency matrix of Γ . Note that underlying graph of Γ is a 4 −regular graph. HenceD (Γ) = diag (4, 4,

· · · , 4). Suppose L(Γ) = D(Γ)

−C s (Γ) then L(Γ) is circulant matrix and its

rst row is [4, −1, −1, · · · , 0, 1, 1]. This implies that the eigenvalues of L(Γ) are

µk = 4 −ωk −ω2k + ω− 2k + ω− k = 4 −2i(sin 2kπ

n + sin

4kπn

), k = 0 , 1, · · · , n −1,

where ω = e2 πi

n and i2 = −1. It is clear that

µn − k = 4 + 2 i(sin 2kπ

n + sin

4kπn

) and µk + µn − k = 8




for k = 1 , 2, · · · ,n −1

2 . So

µ2k + µ2

n − k = 64 −2µk µn − k

= 64

−2(4

−2i(sin

2kπ

n + sin

4kπ

n ))(4 + 2 i(sin

2kπ

n + sin

4kπ

n ))

= 64 −2(16 + 4(sin 2kπ

n + sin

4kπn

)2 )

= 24 −8cos 2kπ

n + 4 cos

4kπn −8cos

6kπn

+ 4cos 8kπ

n

for k = 1 , 2, · · · ,n −1

2 . Let n ≡ 1(mod2), then

α(Γ) =n − 1

k=0

µ2k

= µ20 +

n − 1

k=1 λ2k

= 16 +

n − 12

k=1

(µ2k + µ2

n − k )

= 16 +

n − 12

k=1

(24 −8cos 2kπ

n + 4cos

4kπn −8cos

6kπn

+ 4 cos 8kπ

n )

= 4(3 n + 1) −8

n − 12

k=1

cos 2kπ

n + 4

n − 12

k=1

cos 4kπ

n −8

n − 12

k=1

cos 6kπ

n + 4

n − 12

k=1

cos 8kπ

n

= 4(3 n + 1) −8

n − 12

k=1cos 2kπn + 4

n − 12

k=1cos 2kπn −8

n − 12

k=1cos 6kπn + 4

n − 12

k=1cos 2kπn

(using Lemma2 .11(i), 2.13(vii ))

= 4(3 n + 1) −8

n − 12

k=1

cos 6kπ

n ≤ 4(3n + 1) −8

n − 12

k=1|cos

6kπn |. (2.1)

(i) If n ≡ 3(mod6) then using Lemma 2.13(i) in above inequelity, we get

α(Γ) ≤ 4(3n + 1) −8(32

csc 3π2n −

12

) = 4(3 n + 2) −12 csc 3π2n

.

This completes the proof of ( i).

(ii ) If n ≡ 1 or 5(mod6) and using Lemma 2.13( ii ) and ( iii ), we get

α(Γ) ≤ 4(3n + 1) −8

n − 12

k=1|cos

2kπn |

= 4(3 n + 1) −8(12

csc π2n −

12

) = 4(3 n + 2) −4csc π2n

.




Let n ≡ 0(mod2). Then

α(Γ) =n − 1

k=0

µ2k = µ2

0 + µ2n2

+n − 1

k=1 ,k = n2

µ2k = 16 + 16 +

n − 22

k=1

(µ2k + µ2

n − k )

= 32 +

n − 22

k=1

(24 −8cos 2kπ

n + 4 cos

4kπn −8cos

6kπn

+ 4cos 8kπ

n )

= 4(3 n + 2) −8

n − 22

k=1

cos 2kπ

n + 4

n − 22

k=1

cos 4kπ

n −8

n − 22

k=1

cos 6kπ

n + 4

n − 22

k=1

cos 8kπ

n . (2.2)

If n ≡ 2(mod 4), then employing Lemma 2 .13(viii) in (2.2), we get

α(Γ) = 4(3 n + 2) −8

n − 22

k=1

cos 2kπ

n + 4

n − 22

k=1

cos 4kπ

n −8

n − 22

k=1

cos 6kπ

n + 4

n − 22

k=1

cos 4kπ

n

= 4(3 n + 2) −8

n − 22

k=1

cos 2kπ

n + 8

n − 22

k=1

cos 4kπ

n −8

n − 22

k=1

cos 6kπ

n . (2.3)

(iii ) If n ≡ 2(mod4) and n ≡ 0(mod 6), then using Lemma 2.13( iv) in (2.3) we deducethat

α(Γ) ≤ 4(3n + 2) + 8

n − 22

k=1|cos

2kπn |+ 8

n − 22

k=1|cos

4kπn |

= 4(3 n + 2) + 16(csc πn −1) = 4(3 n −2) + 16 csc

πn

by using Lemma 2.4( ii ) and 2.11( ii ).(iv) If n ≡ 2(mod4) and n ≡ 2 or 4(mod6), then using Lemma 2.13( v) and ( vi) in (2.3) we

see that

α(Γ) = 4(3 n + 2) −8

n − 22

k=1

cos 2kπ

n + 8

n − 22

k=1

cos 4kπ

n −8

n − 22

k=1

cos 2kπ

n

≤ 4(3n + 2) + 16

n − 22

k=1|cos

2kπn |+ 8

n − 22

k=1|cos

4kπn |

≤ 4(3n + 2) + 24(csc πn −1) = 4(3 n −2) + 24csc

πn

.

Similarly we can prove ( v) to ( x).

We give few interesting results on the skew energy of Cayley digraphs on dihedral groupsD2n .

Theorem 2.15 Let D2n = a, b|a2 = bn = 1 , a − 1ba = b− 1 the dihedral group of order 2n and Γ = Cay (D2n , S ) be a Cayley digraph on D2n with respect to S = {bi}, 1 ≤ i ≤ n −1 , and






(i) α(Γ) ≤ 8(3n + 2) −24csc 3π2n if n ≡ 3(mod6).

(ii ) α(Γ) ≤ 8(3n + 2) −8csc π2n if n ≡ 1 or 5(mod6).

(iii ) α(Γ) ≤ 8(3n −2) + 32csc πn if n ≡ 2(mod4) and n ≡ 0(mod6).

(iv) α(Γ) ≤ 8(3n −2) + 48csc πn if n ≡ 2(mod4)and n ≡ 2 or 4(mod6).

(v) α(Γ) ≤ 8(3n −2) + 16cot πn + 16 csc 2π


(vi) α(Γ) ≤ 8(3n −4) + 32cot πn + 16 csc 2π


(vii ) α(Γ) ≤ 8(3n −2) + 16cot πn + 32 cot 4π

n if n ≡ 0(mod16), and n ≡ 0(mod6).

(viii ) α(Γ) ≤ 4(n −8) + 32cot πn + 32cot 4π


(ix ) α(Γ) ≤ 8(3n −2) + 16cot πn + 32 csc 4π


(x) α(Γ)

≤ 8(3n

−4) + 32cot π

n + 32 csc 4πn if n

≡ 8(mod16) and n

≡ 2 or 4(mod6).

Proof The proof of Theorem 2 .18 directly follows from the denition of dihedral groupand Theorem 2 .14.

References

[1] C.Adiga,and R.Balakrishnan, and Wasin So, The skew energy of a digraph, Linear Algebra and its Applications , 432 (2010) 1825-1835.

[2] Andries E.Brouwer, and Willem H.Haemers, Spectra of Graphs , Springer, 2011.[3] Norman Biggs, Algebraic Graph Theory , Cambridge University press,1974.

[4] Gui-Xian Tian, On the skew energy of orientations of hypercubes, Linear Algebra and its Applications , 435 (2011) 2140-2149.

[5] C.Godsil and G.Royle, Algebraic Graph Theory, Springer-Verlag, New York, 2001.[6] V.Nikiforov, The energy of graphs and matrices, J. Math. Anal. Appl. , 320(2007), 1472-

1475.[7] I.Gutman, The energy of a graph, Ber. Math-Satist. Sekt. Forschungsz. Graz , 103(1978),

1-22.




Equivalence of Kropina and Projective Change of

Finsler Metric

H.S.Shukla and O.P.Pandey

(Department of Mathematics & Statistics, DDU Gorakhpur University,Gorakhpur, India)

B.N.Prasad

(C-10, Surajkund Avas Vikas Colony, Gorakhpur, India)

E-mail: [email protected], [email protected], baijnath [email protected]

Abstract : A change of Finsler metric L(x, y ) → L(x, y ) is called Kropina change if L (x, y ) =

L2

β , where β (x, y ) = bi (x ) y i is a one-form on a smooth manifold M n . The

change L → L is called projective change if every geodesic of one space is transformed to ageodesic of the other. The purpose of the present paper is to nd the necessary and sufficientcondition under which a Kropina change becomes a projective change.

Key Words : Kropina change, projective change, Finsler space.

AMS(2010) : 53C60, 53B40

§1. Preliminaries

Let F n = ( M n , L) be a Finsler space equipped with the fundamental function L(x, y ) on the

smooth manifold M n . Let β = bi (x)yi be a one-form on the manifold M n , then L → L2

β is called

Kropina change of Finsler metric [5]. If we write L = L2

β and F

n= ( M n , L), then the Finsler

space F n

is said to be obtained from F n by Kropina change. The quantities corresponding toF

nare denoted by putting bar on those quantities.The fundamental metric tensor gij , the normalized element of support li and angular metric

tensor h ij of F n are given by

gij =

1

2

∂ 2L2

∂y i ∂y j , li =

∂L

∂y i and hij = L

∂ 2L

∂y i ∂y j = gij −li lj .

We shall denote the partial derivative with respect to x i and yi by ∂ i and ∂ i respectivelyand write

L i = ∂ i L, L ij = ∂ i ∂ j L, L ijk = ∂ i ∂ j ∂ k L.




78 H.S.Shukla, O.P.Pandey and B.N.Prasad

ThusL i = li , L− 1 h ij = Lij

The geodesic of F n are given by the system of differential equations

d2x

i

ds2 + 2Gi x, dxds

= 0 ,

where G i (x, y ) are positively homogeneous of degree two in yi and is given by

2G i = gij (yr ∂ j ∂ r F −∂ j F ), F = L2

2

where gij are the inverse of gij .The well known Berwald connection BΓ = {G i

jk , G ij } of a Finsler space is constructed from

the quantity G i appearing in the equation of geodesic and is given by [6]

G ij = ∂ j Gi , Gi

jk = ∂ k Gij .

The Cartan’s connection C Γ = {F ijk , G ij , C ijk } is constructed from the metric function L

by the following ve axioms [6]:

(i) gij | k = 0; ( ii ) gij |k = 0; ( iii ) F ijk = F ikj ; (iv) F i0k = Gik ; (v) C ijk = C ikj .

where |k and |k denote h and v−covariant derivatives with respect to C Γ. It is clear thatthe h−covariant derivative of L with respect to B Γ and C Γ are same and vanishes identically.Furthermore the h−covariant derivatives of L i , L ij with respect to C Γ are also zero.

We denote2r ij = bi | j + bj | i , 2s ij = bi | j −bj | i .

§2. Kropina Change of Finsler Metric

The Kropina change of Finsler metric L is given by

(2.1) L = L2

β , where β (x, y ) = bi (x) yi .

We may put

(2.2) Gi

= Gi + D i .

Then Gij = G i

j + D ij and G

ijk = G i

jk + D ijk , where D i

j = ∂ j D i and D ijk = ∂ k D i

j . The tensorsD i , D i

j and D ijk are positively homogeneous in yi of degree two, one and zero respectively.

To nd D i we deal with equations L ij | k = 0 [2], where L ij | k is the h−covariant derivativeof L ij = hij /L with respect to Cartan’s connection C Γ. Then

(2.3) ∂ k L ij −L ijr Grk −Lrj F rik −L ir F rjk = 0 .

Since ∂ i β = bi , from (2.1), we have










where we have written s0 for s r 0br .Equation (2.13) can be written as

(2.17) −2L2(br D r ) + 4 βL (L r D r ) = −L2r 00 .

The equation (2.16) and (2.17) constitute the system of algebraic equations in Lr D r andbr D r . Solving equations (2.16) and (2.17) for br D r and Lr D r , we get

(2.18) br D r = 12L2b2 [(b2L2 −2β 2)r00 −2βL 2s0]

and

(2.19) Lr D r = − 1

2L2b2 [L3s0 + βLr 00 ].

Contracting (2.15) by gji and re-arranging terms, we obtain

(2.20) D j =2βL (br D r ) −βLr 00

2β 2 lj +

L2r 00 + 2 βL (Lr D r ) −2L2(br D r )

2β 2 bj

− L2

2β s j

0

Putting the values of br D r and Lr D r from equations (2.18) and (2.19) respectively in (2.20),we get

(2.21) D i =βr 00 + L2s0

2b2β bi −

βr 00 + L2s0

b2L2 yi − L2

2β s i

0 , where li = yi

L.

Proposition 2.1 The difference tensor D i = Gi

−G i of Kropina change of Finsler metric is given by (2.21).

§3. Projective Change of Finsler Metric

The Finsler space F n

is said to be projective to Finsler space F n if every geodesic of F n istransformed to a geodesic of F

n. Thus the change L → L is projective if G

i= Gi + P (x, y )yi ,

where P (x, y) is a homogeneous scalar function of degree one in yi , called projective factor [4].Thus from (2.2) it follows that L → L is projective iff D i = P yi . Now we consider that

the Kropina change L → L = L2

β is projective. Then from equation (2.21), we have

(3.1) P yi =βr 00 + L2s0

2b2β bi −

βr 00 + L2s0

b2L2 yi − L2

2β s i

0 .

Contracting (3.1) by yi (= gij yj ) and using the fact that s i0yi = 0 and yi yi = L2 , we get

(3.2) P = − 1

2b2L2 (βr 00 + L2s0).

Putting the value of P from (3.2) in (3.1), we get

(3.3)βr 00 + L2s0

2b2L2 yi =βr 00 + L2s0

2b2β bi −

L2

2β s i

0 .



Equivalence of Kropina and Projective Change of Finsler Metric 83

Transvecting (3.3) by bi , we get

(3.4) r00 = −βs 0 , where =

β L

2

−b2 = 0 .

Substituting the value of r 00 from (3.4) in (3.2), we get

(3.5) P = 12

s0 .

Eliminating P and r 00 from (3.5), (3.4) and (3.2), we get

(3.6) si0 =

β L2 yi −bi s0 .

The equations (3.4) and (3.6) give the necessary conditions under which a Kropina changebecomes a projective change.

Conversely, if conditions (3.4) and (3.6) are satised, then putting these conditions in(2.21), we get

D i = s0

2 yi i.e. D i = P yi , where P =

s0

2.

Thus F n

is projective to F n .

Theorem 3.1 The Kropina change of a Finsler space is projective if and only if (3.4) and

(3.6) hold and then the projective factor P is given by P = s02 , where = β

L

2

−b2 .

§4. A Particular Case

Let us assume that L is a metric of a Riemannian space i.e. L = a ij (x)yi yj = α. Then

L = α2

β which is the metric of Kropina space. In this case bi | j = bi ;j where ; j denotes the

covariant derivative with respect to Christoffel symbols constructed from Riemannian metric α.Thus r ij and s ij are functions of coordinates only and in view of theorem (3.1) it follows that

the Riemannian space is projective to Kropina space iff r00 = −β

s0 and si0 =

β α 2 yi −bi s 0 ,

where =β α

2

−b2 = 0. These equations may be written as

(4.1) (a) r 00 β 2 = α2(b2r 00 −βs 0); (b) s i0(β 2 −b2α 2) = ( β 2 yi −α 2bi )s0 .

From (4.1)(a), it follows that if α2

≡ o(mod β ) i.e. β is not a factor of α2, then there exists

a scalar function f (x) such that

(4.2) (a) b2r 00 −βs 0 = β 2 f (x); (b) r 00 = α2 f (x).

From (4.2)(b), we get r ij = f (x)a ij and therefore (4.2)(a) reduces to

βs 0 = ( b2α 2 −β 2)f (x).




This equation may be written as

bi s j + bj s i = 2( b2a ij −bi bj )f (x)

which after contraction with bj gives b2s i = 0. If b2 = 0 then we get s i = 0, i.e. sij = 0 .Hence equation (4.1) holds identically and (4.2)(a) and (b) give

(b2α2 −β 2 )f (x) = 0 i .e. f (x) = 0 as b2α 2 −β 2 = 0 .

Thus r 00 = 0, i.e. r ij = 0. Hence bi ;j = 0, i.e. the pair ( α, β ) is parallel pair.Conversely, if bi ;j = 0, the equation (4.1)(a) and (4.1)(b) hold identically. Thus we get the

following theorem which has been proved in [1], [7].

Theorem 4.1 The Riemannian space with metric α is projective to a Kropina space with

metric α2

β iff the (α, β ) is parallel pair.

References

[1] Bacso S. and Matsumoto M., Projective changes between Finsler spaces with ( α, β )−metric,Tensor (N. S.) , 55 (1994), 252-257.

[2] Matsumoto M., On Finsler space with Randers metric and special forms of importanttensors, J. Math. Kyoto Univ. , 14 (1974), 477-498.

[3] Matsumoto M., Foundations of Finsler Geometry and Special Finsler Spaces , KaiseishaPress, Otsu, Japan, 1986.

[4] Matsumoto M., Theory of Finsler spaces with ( α, β )−metric, Rep. Math. Phy. , 31 (1992),43-83.

[5] Park H. S. and Lee I. Y., Projective changes between a Finsler space with ( α, β )−metricand the associated Riemannian space, Tensor (N. S.) , 60 (1998), 327-331.[6] Park H.S. and Lee I.Y., The Randers changes of Finsler spaces with ( α, β )−metrics of

Douglas type, J. Korean Math. Soc. , 38 (2001), 503-521.[7] Singh U.P., Prasad B.N. and Kumari Bindu, On Kropina change of Finsler metric, Tensor

(N. S.) , 63 (2003), 181-188.




Geometric Mean Labeling

Of Graphs Obtained from Some Graph Operations

A.Durai Baskar, S.Arockiaraj and B.Rajendran

Department of Mathematics, Mepco Schlenk Engineering College

Mepco Engineering College (PO)-626005, Sivakasi, Tamil Nadu, India

E-mail: [email protected], sarockiaraj [email protected], [email protected]

Abstract : A function f is called a geometric mean labeling of a graph G(V, E ) if f :V (G ) → {1, 2, 3, . . . , q + 1 } is injective and the induced function f : E (G ) → {1, 2, 3, . . . , q }dened as

f (uv ) = f (u )f (v) , uv E (G ) ,

is bijective. A graph that admits a geometric mean labeling is called a geometric meangraph. In this paper, we have discussed the geometric meanness of graphs obtained fromsome graph operations.

Key Words : Labeling, geometric mean labeling, geometric mean graph.

AMS(2010) : 05C78

§1. Introduction

Throughout this paper, by a graph we mean a nite, undirected and simple graph. Let G(V, E )be a graph with p vertices and q edges. For notations and terminology, we follow [3]. For adetailed survey on graph labeling, we refer [2].

Cycle on n vertices is denoted by C n and a path on n vertices is denoted by P n . A tree T is a connected acyclic graph. Square of a graph G, denoted by G2 , has the vertex set as in Gand two vertices are adjacent in G2 if they are at a distance either 1 or 2 apart in G. A graphobtained from a path of length m by replacing each edge by C n is called as mC n -snake, form ≥ 1 ad n ≥ 3.

The total graph T (G) of a graph G is the graph whose vertex set is V (G) E (G) and twovertices are adjacent if and only if either they are adjacent vertices of G or adjacent edges of G or one is a vertex of G and the other one is an edge incident on it. The graph TadpolesT (n, k ) is obtained by identifying a vertex of the cycle C n to an end vertex of the path P k . TheH -graph is obtained from two paths u1 , u2 , . . . , u n and v1 , v2 , · · · , vn of equal length by joiningan edge u n +1

2v n +1

2when n is odd and u n +2

2v n

2when n is even. An arbitrary supersubdivision

P (m1 , m 2 , · · · , mn − 1) of a path P n is a graph obtained by replacing each ith edge of P n byidentifying its end vertices of the edge with a partition of K 2,m i having 2 elements, where m i is




86 A.Durai Baskar, S.Arockiaraj and B.Rajendran

any positive integer. G K 1 is the graph obtained from G by attaching a new pendant vertexto each vertex of G.

The study of graceful graphs and graceful labeling methods was rst introduced by Rosa[5]. The concept of mean labeling was rst introduced by S.Somasundaram and R.Ponraj [6]

and it was developed in [4,7]. S.K.Vaidya et al. [11] have discussed the mean labeling in thecontext of path union of cycle and the arbitrary supersubdivision of the path P n . S.K.Vaidyaet al. [8-10] have discussed the mean labeling in the context of some graph operations. In [1],A.Durai Baskar et al. introduced geometric mean labeling of graph.

A function f is called a geometric mean labeling of a graph G(V, E ) if f : V (G) →{1, 2, 3, · · · , q + 1} is injective and the induced function f : E (G) → {1, 2, 3, · · · , q } dened as

f (uv) = f (u)f (v) , uv E (G),

is bijective. A graph that admits a geometric mean labeling is called a geometric mean graph.

In this paper we have obtained the geometric meanness of the graphs, union of two cyclesC m and C n , union of the cycle C m and a path P n , P 2

n , mC n -snake for m ≥ 1 and n ≥ 3, thetotal graph T (P n ) of P n , the Tadpoles T (n, k ), the graph obtained by identifying a vertex of any two cycles C m and C n , the graph obtained by identifying an edge of any two cycles C mand C n , the graph obtained by joining any two cycles C m and C n by a path P k , the H -graphand the arbitrary supersubdivision of a path P (1, 2, · · · , n −1).

§2. Main Results

Theorem 2.1 Union of any two cycles C m and C n is a geometric mean graph.

Proof Let u1 , u2 , · · · , um and v1 , v2 , · · · , vn be the vertices of the cycles C m and C n re-spectively. We dene f : V (C m C n ) → {1, 2, 3, · · · , m + n + 1} as follows:

f (u i ) =i if 1 ≤ i ≤ √ m + 2 −1

i + 1 if √ m + 2 ≤ i ≤ m −1,

f (um ) = m + 2 and

f (vi ) =

m + n + 3 −2i if 1 ≤ i ≤ n2

m + 1 if i = n2 + 1

m −n + 2 i if n2 + 2 ≤ i ≤ n.

The induced edge labeling is as follows:

f (u i u i +1 ) =i if 1 ≤ i ≤ √ m + 2 −1

i + 1 if √ m + 2 ≤ i ≤ m −1,



Geometric Mean Labeling of Graphs Obtained from Some Graph Operations 87

f (u, u m ) = √ m + 2 ,

f (vi vi+1 ) =

m + n + 1 −2i if 1 ≤ i ≤ n2

m + 1 if i = n2 + 1 and n is odd

m + 2 if i =n2 + 1 and n is even

m −n + 2 i if n2 + 2 ≤ i ≤ n −1

and f (v1 vn ) = m + n.

Hence, f is a geometric mean labeling of the graph C m C n . Thus the graph C m C n isa geometric mean graph, for any m, n ≥ 3.

A geometric mean labeling of C 7 C 10 is shown in Fig.1.

u1

u2

u3

u4

u5

u6

u7

13

9

7

7

6

6

55

4

4

2

2v1

v2

v3

v4

v5

v6

v7

v8

v9

v10

1816 16

14141212

10

10

88

911

11

13

13

1515

17

Fig. 1

The graph C m nT,n ≥ 2 cannot be a geometric mean graph. But the graph C m T maybe a geometric mean graph.

Theorem 2.2 The graph C m P n is a geometric mean graph.

Proof Let u1 , u2 , · · · , um and v1 , v2 , · · · , vn be the vertices of the cycle C m and the pathP n respectively. We dene f : V (C m P n ) → {1, 2, 3, · · · , m + n} as follows:

f (u i ) =

m + n + 2 −2i if 1 ≤ i ≤ m2n if i =

m2

+ 1

n −m −1 + 2 i if m2

+ 2 ≤ i ≤ m,

f (vi ) = i, for 1 ≤ i ≤ n −1 and

f (vn ) = n + 1 .






f (vi ) =

m + n + 2 −2i if 1 ≤ i ≤m2

n if i =m2

+ 1

n −m −1 + 2 i if m2

+ 2 ≤ i ≤ m.


f (u i u i+1 ) = i + 1 , for 1 ≤ i ≤ n −2,

f (u2un ) = 1 ,

f (vi vi+1 ) =

m + n −2i if 1 ≤ i ≤m2

n if i =m2

+ 1 and m is odd

n + 1 if i =m2

+ 1 and m is even

n

−m

−1 + 2 i if

m

2+ 2

≤ i

≤ m

−1

and

f (v1 vm ) = m + n −1.

Hence f is a geometric mean labeling of T n C m . Thus the graph T n C m is a geometric meangraph, for n ≥ 2 and m ≥ 3.

A geometric mean labeling of T 7 C 6 is as shown in Fig.3.

2 2 3 1 1

344

5

5

6

6

8

13

11

11

9

97

78

10

10

1212

Fig. 3

Theorem 2.4 P 2n is a geometric mean graph, for n ≥ 3.

Proof Let v1 , v2 , · · · , vn be the vertices of the path P n . We dene f : V (P 2n ) → {1, 2, 3, · · · , 2(n−1)} as follows:

f (vi ) = 2 i −1, for 1 ≤ i ≤ n −1 and

f (vn ) = 2( n −1).






Case 2 n = 4 .

Let v( i )1 , v( i )

2 , v( i )3 and v( i )

4 be the vertices of the ith copy of the cycle C 4 , for 1 ≤ i ≤ m.The mC 4-snake G is obtained by identifying v( i )

4 and v( i+1)1 , for 1 ≤ i ≤ m − 1. We dene

f : V (G)

→ {1, 2, 3,

· · · , 4m + 1

} as follows:

f (v( i )1 ) = 4 i −3, for 1 ≤ i ≤ m,

f (v( i )2 ) = 4 i −1, for 1 ≤ i ≤ m,

f (v( i )3 ) = 4 i, for 1 ≤ i ≤ m and

f (v( i )4 ) = 4 i + 1 , for 1 ≤ i ≤ m.


f (v( i )1 v( i )

2 ) = 4 i −3, for 1 ≤ i ≤ m,

f (v( i )2 v( i )

3 ) = 4 i

−1, for 1

≤ i

≤ m

f (v( i )3 v( i )

4 ) = 4 i, for 1 ≤ i ≤ m and

f (v( i )1 v( i )

4 ) = 4 i −2, for 1 ≤ i ≤ m.

Hence, f is a geometric mean labeling of the graph mC 4-snake.

A geometric mean labeling of 5 C 4-snake is shown in Fig.6.

1 2 5 6 9 10 13 14 17 18 21

1

3 3 4

4 5

7 7 8

8 9

11 11 12

12 13

15 15 16

16 17

19 19 20

20

Fig. 6

Theorem 2.6 T (P n ) is a geometric mean graph, for n ≥ 2.

Proof Let V (P n ) = {v1 , v2 , · · · , vn } and E (P n ) = {ei = vi vi+1 ; 1 ≤ i ≤ n − 1} be thevertex set and edge set of the path P n . Then

V (T (P n )) = {v1 , v2 , . . . , vn , e1 , e2 , · · · , en − 1} and

E (T (P n )) =

{vi vi +1 , ei vi , ei vi+1 ; 1

≤ i

≤ n

−1

} {ei ei +1 ; 1

≤ i

≤ n

−2

}.

We dene f : V (T (P n )) → {1, 2, 3, · · · , 4(n −1)} as follows:

f (vi ) = 4 i −3, for 1 ≤ i ≤ n −1,

f (vn ) = 4 n −4 and

f (ei ) = 4 i −1, for 1 ≤ i ≤ n −1.





f (vi vi +1 ) = 4 i −2, for 1 ≤ i ≤ n −1,

f (ei ei +1 ) = 4 i, for 1 ≤ i ≤ n −2,

f (ei vi ) = 4 i −3, for 1 ≤ i ≤ n −1 and

f (ei vi +1 ) = 4 i −1, for 1 ≤ i ≤ n −1.

Hence, f is a geometric mean labeling of the graph T (P n ). Thus the graph T (P n ) is a geometricmean graph, for n ≥ 2.

A geometric mean labeling of T (P 5) is shown in Fig.7.

v1 v2 v3 v4 v5

e1 e2 e3 e4

1

1

3

3

25

5

7

7

9

9

11

11

13

13

14

15

15

16106

4 8 12

Fig. 7

Theorem 2.7 Tadpoles T (n, k ) is a geometric mean graph.

Proof Let u1 , u2 , · · · , un and v1 , v2 , · · · , vk be the vertices of the cycle C n and the path P krespectively. Let T (n, k ) be the graph obtained by identifying the vertex un of the cycle C n tothe end vertex v1 of the path P k . We dene f : V (T (n, k ))

→ {1, 2, 3,

· · · , n + k

} as follows:

f (u i ) =i if 1 ≤ i ≤ √ n + 1 −1

i + 1 if √ n + 1 ≤ i ≤ nand

f (vi ) = n + i, for 2 ≤ i ≤ k.


f (u i u i+1 ) =i if 1 ≤ i ≤ √ n + 1 −1

i + 1 if √ n + 1 ≤ i ≤ n −1,

f (u1un ) = √ n + 1 andf (vi vi+1 ) = n + i, for 1 ≤ i ≤ k −1.

Hence, f is a geometric mean labeling of the graph T (n, k ). Thus the graph T (n, k ) is a geometricmean graph.

A geometric mean labeling of the Tadpoles T (7, 5) is shown in Fig.8.




u2

u3

u4

u5

u6

u7

u1

31

1

2

7

7

66

5

5

4

4

v1 v2 v3 v4 v58 8 9 9 10 10 11 11 12

3

Fig. 8

Theorem 2.8 The graph obtained by identifying a vertex of any two cycles C m and C n is a geometric mean graph.

Proof Let u1 , u2 , · · · , um and v1 , v2 , · · · , vn be the vertices of the cycles C m and C n re-spectively. Let G be the resultant graph obtained by identifying the vertex um of the cycle C mto the vertex vn of the cycle C n . We dene f : V (G) → {1, 2, 3, · · · , m + n + 1 } as follows:

f (u i ) =i if 1 ≤ i ≤ √ m + 1 −1

i + 1 if √ m + 1 ≤ i ≤ mand

f (vi ) =m + 1 + i if 1 ≤ i ≤ (m + 1)( m + n + 1) −m −2

m + 2 + i if

(m + 1)( m + n + 1) −m −1 ≤ i ≤ n −1.


f (u i u i +1 ) =i if 1 ≤ i ≤ √ m + 1 −1,

i + 1 if √ m + 1 ≤ i ≤ m −1,

f (vi vi +1 ) =m + 1 + i if 1 ≤ i ≤ (m + 1)( m + n + 1) −m −2,

m + 2 + i if (m + 1)( m + n + 1) −m −1 ≤ i ≤ n −2,

f (u1um ) = √ m + 1 ,

f (vn − 1vn ) =

(m + 1)( m + n + 1) and

f (v1vn ) = m + 1 .

Hence, f is a geometric mean labeling of the graph G. Thus the resultant graph G is a geometricmean graph.

A geometric mean labeling of the graph G obtained by identifying a vertex of the cyclesC 8 and C 12 , is shown in Fig.9.






v3v4

v5

v6

v7

v8v9

v10

v11

v1

v2u1u2

u3

u4

u6u7

u8

22

4

4

5

6

77 8

89

9

3

11

u10

u9

u5

5

6

10v12

v13

1212

13 13 14

1416

161717

1818

1919

202021

2122

22231510

1111

Fig. 10

Theorem 2.10 The graph obtained by joining any two cycles C m and C n by a path P k is a

geometric mean graph.Proof Let G be a graph obtained by joining any two cycles C m and C n by a path P k .

Let u1 , u 2 , · · · , um and v1 , v2 , · · · , vn be the vertices of the cycles C m and C n respectively.Let w1 , w2 , . . . , wk be the vertices of the path P k with um = w1 and wk = vn . We denef : V (G) → {1, 2, 3, · · · , m + k + n} as follows:

f (u i ) =i if 1 ≤ i ≤ √ m + 1 −1

i + 1 if √ m + 1 ≤ i ≤ m,

f (wi ) = m + i, for 2 ≤ i ≤ k and

f (vi ) =m + k + i if 1

≤ i

≤ (m + k)(m + k + n)

−m

−k

−1

m + k + 1 + i if (m + k)(m + k + n) −m −k ≤ i ≤ n −1.


f (u i u i +1 ) =i if 1 ≤ i ≤ √ m + 1 −1

i + 1 if √ m + 1 ≤ i ≤ m −1,

f (wi wi +1 ) = m + i, for 1 ≤ i ≤ k −1,

f (vi vi +1 ) =m + k + i if 1 ≤ i ≤ (m + k)(m + k + n) −m −k −1

m + k + 1 + i if

(m + k)(m + k + n) −m −k ≤ i ≤ n −2,

f (u1um ) = √ m + 1 ,

f (vn vn − 1) = (m + k)(m + k + n) and

f (v1vn ) = m + k.

Hence, f is a geometric mean labeling of the graph G. Thus the resultant graph G is a geometricmean graph.




A geometric mean labeling of the graph G obtained by joining two cycles C 7 and C 10 bya path P 4 , is shown in Fig.11.

v7v8

v9

v10

v2v3

v4

2020

21

15

1313 14

1416

18

1919

v111

12

v6

v5

u1

u2

u4

u5

u6

u7

113

3

4

4

5

5

u3

66

7

7

8

2

w1 w2 w3 w4

12

11101099818

17

17

16

Fig. 11

Theorem 2.11 Any H -graph G is a geometric mean graph.

Proof Let u1 , u2 , · · · , un and v1 , v2 , · · · , vn be the vertices on the paths of length n in G.

Case 1 n is odd.

We dene f : V (G) → {1, 2, 3, · · · , 2n} as follows:

f (u i ) = i, for 1 ≤ i ≤ n and

f (vi ) =

n + 2 i if 1 ≤ i ≤n2

n + 2 i −1 if i =n2

+ 1

3n + 1 −2i if n2 + 2 ≤ i ≤ n.


f (u i u i+1 ) = i, for 1 ≤ i ≤ n −1,

f (u i vi ) = n, for i =n2

+ 1 and

f (vi vi+1 ) =n + 2 i if 1 ≤ i ≤

n2

3n −1 −2i if n2

+ 1 ≤ i ≤ n −1.

Case 2 n is even.We dene f : V (G) → {1, 2, 3, · · · , 2n} as follows:

f (u i ) = i, for 1 ≤ i ≤ n and

f (vi ) =n + 2 i if 1 ≤ i ≤

n2

3n + 1 −2i if n2

+ 1 ≤ i ≤ n.





f (u i u i+1 ) = i, for 1 ≤ i ≤ n −1,

f (u i +1 vi ) = n, for i =n2

and

f (vi vi+1 ) =n + 2 i if 1 ≤ i ≤ n

2 −1

3n −1 −2i if n2 ≤ i ≤ n −1.

Hence, H -graph admits a geometric mean labeling.

A geometric mean labeling of H -graphs G1 and G2 are shown in Fig.12.u1

u2

u3

u4

u5

u6

u7

u8

u9

v1

v2

v3

v4

v5

v6

v7

v8

v9

1

1

22

33

44

55

66

77

8

89 10

10

121214

1416

1618

1717

1515

1313

1111 u1

u2

u3

u4

u5

u6

u7

u8

u9

u10 v10

v9

v8

v7

v6

v5

v4

v3

v2

v1

10

1122

33

44

55

66

77

88

99

10 1111

131315

15171719

1920

1818

1616

1414

1212

G1 G2Fig. 12

Theorem 2.12 For any n ≥ 2, P (1, 2, 3, · · · , n −1) is a geometric mean graph.

Proof Let v1 , v2 , · · · , vn be the vertices of the path P n and let uij be the vertices of the partition of K 2,m i with cardinality mi , 1 ≤ i ≤ n − 1 and 1 ≤ j ≤ mi . We dene f :V (P (1, 2, · · · , n −1)) → {1, 2, 3, . . . , n (n −1) + 1 } as follows:

f (vi ) = i(i −1) + 1 , for 1 ≤ i ≤ n andf (u ij ) = i(i −1) + 2 j, for 1 ≤ j ≤ i and 1 ≤ i ≤ n −1.


f (vi u ij ) = i(i −1) + j, for 1 ≤ j ≤ i and 1 ≤ i ≤ n −1

f (u ij vi +1 ) = i2 + j, for 1 ≤ j ≤ i and 1 ≤ i ≤ n −1.




Hence, f is a geometric mean labeling of the graph P (1, 2, · · · , n−1). Thus the graph P (1, 2, · · · , n−1) is a geometric mean graph.

A geometric mean labeling of P (1, 2, 3, 4, 5) is shown in Fig.13.

u11 u21 u31

u41

u42

v1 v2 v3 v4 v5

u22 u33 u43

u44

1

12 2

3

34

5

78

78

10

u32

10 11

46

6 912

12

13

1416

18

21

19

1815

16 20

20

1713

14

Fig. 13

References

[1] A.Durai Baskar, S.Arockiaraj and B.Rajendran, Geometric Mean Graphs (Communicated).[2] J.A.Gallian, A dynamic survey of graph labeling, The Electronic Journal of Combinatorics ,

17 (2011).[3] F.Harary, Graph Theory , Addison Wesely, Reading Mass., 1972.[4] R.Ponraj and S.Somasundaram, Further results on mean graphs, Proceedings of Sacoefer-

ence , (2005), 443-448.[5] A.Rosa, On certain valuation of the vertices of graph, International Symposium, Rome,

July 1966, Gordon and Breach, N.Y. and Dunod Paris (1967), 349-355.[6] S.Somasundaram and R.Ponraj, Mean labeling of graphs, National Academy Science Letter ,

26 (2003), 210-213.[7] S.Somasundaram and R.Ponraj, Some results on mean graphs, Pure and Applied Mathe-

matika Sciences , 58 (2003), 29-35.[8] S.K.Vaidya and Lekha Bijukumar, Mean labeling in the context of some graph operations,

International Journal of Algorithms, Computing and Mathematics , 3(2010).[9] S.K.Vaidya and Lekha Bijukumar, Some netw families of mean graphs, Journal of Mathe-

matics Research , 2(3) (2010).[10] S.K.Vaidya and Lekha Bijukumar, Mean labeling for some new families of graphs, Journal of Pure and Applied Sciences , 8 (2010), 115-116.

[11] S.K.Vaidya and K.K.Kanani, Some new mean graphs, International Journal of Information Science and Computer Mathematics , 1(1) (2010), 73-80.




4-Ordered Hamiltonicity of the

Complete Expansion Graphs of Cayley Graphs

Lian Ying, A Yongga, Fang Xiang and Sarula

College of Mathematics Science, Inner Mongolia Normal University, Hohhot, 010022, P.R.China


Abstract : In this paper, we prove that the Complete expansion graph And (k) (k ≥ 6) is4-ordered hamiltonian graph by the method of classication discuss.

Key Words : Andr asfai graph, complete expansion graph, k-ordered hamiltonian graph.

AMS(2010) : 05C38 05C45 05C70

§1. Introduction

All graphs considered in this paper are nite, simple and undirected. Let C be a cycle withgiven orientation in graph X , C (C = C ) with anticlockwise direction and C with clockwisedirection. If x V (C ), then we use x+ to denote the successor of x on C and x− to denote itspredecessor. Use C [x, y ] denote (x, y )-path on C ; C (x, y ) denote ( x, y )-path missing x, y on C .Any undened notation follows that of [1, 2].

Denition 1.1([1]) Let G be a group and let C be a subset of G that is closed under taking inverses and does not contain the identity, then the Cayley graph X (G, C ) is the graph with vertex set G and edge set E (X (G, C )) = {gh : hg− 1 C }.

For a Cayley graph G, it may not be a hamiltonian graph, but a Cayley graph of Abeliangroup is a hamiltonian graph. And(k) is a family of Cayley graph, which is named by theHungarian mathematician Andr´ asfai, it is a k-regular graph with the order n = 3k −1 and itis a hamiltonian graph.

Denition 1.2([1]) For any integer k ≥ 1, let G = Z 3k− 1 denote the additive group of integer modulo 3k−1 and let C be the subset of Z 3k− 1 consisting of the elements congruent to 1 modulo

3. Then we denote the Cayley graph X (G, C ) by And (k).

For convenience, we note Z 3k− 1 = {u0 , u1 , . . . , u 3k− 2}. For ui , u j V [And (k)], u i u j if and only if j −i ≡ ±1mod3. The result are directly by the denition of Andr´ asfai graph.

1 Supported by Natural Science Foundation of Inner Mongolia, 2010MS0113; Inner Mongolia Normal Univer-sity Graduate Students’ Research and Innovation Fund. CXJJS11042.




100 Lian Ying, A Yongga, Fang Xiang and Sarula

Lemma 1.3 Let C be any hamiltonian cycle in And(k)(k ≥ 2).

(1) If u, x V (And (k)), u x is a chord of C , then u− x− , u+ x+ .(2) If u,x,y V (And (k)) , u x, u y are two chords of C , then x y+ .

The denition of k-ordered hamiltonian graph was given in 1997 by Lenhard as follows.

Denition 1.4([3]) A hamiltonian graph G of order ν is k-ordered, 2 ≤ k ≤ ν , if for ev-ery sequence (v1 , v2 , . . . , vk ) of k distinct vertices of G, there exists a hamiltonian cycle that encounters (v1 , v2 , . . . , vk ) in this order.

Faudree developed above denition into a k-ordered graph.

Denition 1.5([4]) For a positive integer k, a graph G is k-ordered if for every ordered set of k vertices, there is a cycle that encounters the vertices of the set in the given order.

It has been shown that And(k)(k

≥ 4) is 4-ordered hamiltonian graph by in [5]. The

concept of expansion transformation graph of a graph was given in 2009 by A Yongga at rst.Then an equivalence denition of complete expansion graph was given by her, that is, themethod dened by Cartesian product in [6] as follows.

Denition 1.6([6]) Let G be any graph and L(G) be the line graph of G. Non-trivial component of G L(G) is said complete expansion graph (CEG for short) of G, denoted by (G), said the map be a complete expansion transformation of G.

The proof of main result in this paper is mainly according to the following conclusions.

Theorem 1.7([1]) The Cayley graph X (G, C ) is vertex transitive.

Theorem 1.8([5]) And(k)(k ≥ 4) is 4-ordered hamiltonian graph.

Theorem 1.9([7]) Every even regular graph has a 2-factor.

The notations following is useful throughout the paper. For u V (G), the clique withthe order dG (u) in (G) by u is denoted as (u). All cliques are the cliques in (And (k))determined by the vertices in And(k), that is maximum Clique. For u, v V (G), (u) (v)means there exist x V ( (u)), y V ( (v)), s.t. x y in V ( (G)), edge ( x, y ) is said an edgestretching out from (u). Use G (u ) [x, y ; s, t ] to denote ( x, y )-longest path missing s, t in (u),where x,y,s, t V ( (u)).

§2. Main Results with Proofs

We consider that whether (And (k)) (k ≥ 4) is 4-ordered hamiltonian graph or not in thissection.

Theorem 2.1 (And (k)) ( k ≥ 6) is a 4-ordered hamiltonian graph.



4-Ordered Hamiltonicity of the Complete Expansion Graphs of Cayley Graphs 101

The following lemmas are necessary for the proof of Theorem 2 .1.

Lemma 2.2 For any u V (And (k))( k ≥ 2), x, y N (u), there exists a hamiltonian cycle C in And (k), s.t. ux E (C ) and uy E (C ).

Proof Let C 0 is a hamiltonian cycle u0 u1 u2 . . . u3k− 2 u0 in And (k)(k ≥ 2).For u V (And (k))( k ≥ 2), x, y N (u), then we consider the following cases.

Case 1 x u y on C 0 . Then C = C 0 is that so, since C 0 is a hamiltonian cycle.

Case 2 x u and u y on C 0 or x u and u y on C 0 . If x u and u y on C 0 , thenwe can nd a hamiltonian cycle C in And (k)(k ≥ 2) according to Lemma 1, that is,

C = u x C 0 (x, y − ) y− u− C 0(u− , y) y u;

If x u and u y on C 0 , then we can nd a hamiltonian cycle C in And (k)(k ≥ 2) accordingto Lemma 1 .3, that is,

C = u x C 0 (x, u + ) u+ x+ C 0(x+ , y) y u.

Case 3 x u y on C 0 . Then we can nd a hamiltonian cycle C in And (k)(k ≥ 2) accordingto Lemma 1 .3, that is,

C = u x y+ C 0(y+ , u− ) u− x− C 0(x− , u+ ) u+ C 0[x+ , y] u.

For any u V (And (k))( k ≥ 2), Lemma 2 .2 is true since And (k) is vertex transitive.

Corollary 2.3 For any two edges which stretch out from any Clique, there exists a hamiltonian cycle in (And (k)) containing them.

Lemma 2.4 If k is an odd number, then And(k)(k ≥ 3) can be decomposed into one 1-factor

and k −1

2 2-factors.

Proof 3k − 1 is an even number, since k is an odd number. There exists one 1-factorM in And(k) by the denition of And(k). According to Theorem 1 .9 and the condition of Lemma 2 .4 for integers k ≥ 3, And(k) −E (M ) is a (k −1)-regular graph with a hamiltoniancycle C 1 , And (k) −E (M ) −E (C 1) is a (k −3)-regular graph with a hamiltonian cycle C 2 , · · ·,And (k) −E (M ) −

k − 12

i =1E (C i ) is an empty graph.

Assume k = 2r + 1( r Z + ), since k is an odd number. First we shall prove the result for

r = 1, and then by induction on r . If r = 1 (k = 3), it is easy to see that And (k) −E (M ) isa hamiltonian cycle C 1 by Theorem 1 .9 and the analysis form of Lemma 2 .4 , so the result isclearly true.

Now, we assume that the result is true if r = n(r ≥ 1,k = 2n +1), that is, And (2n + 1) canbe decomposed into one 1-factor and n 2-factors. Considering the case of r = n + 1( k = 2n + 3,we know And (2n +3)( And [2(n +1) +1]) can be decomposed into one 1-factor and n +1 2-factorsaccording to the induction.




Thus, if k is an odd number, then And (k)(k ≥ 3) can be decomposed into one 1-factor andk −1

2 2-factors.

Proof of Theorem 2.1

(And (k)) is a hamiltonian graph, since And(k) is a hamiltonian graph. So there existsa hamiltonian cycle C 0 in (And (k)) and a hamiltonian cycle C 0 ′ in And(k), such that C 0 =

(C 0 ′ ), without loss of generality

C 0 ′ = u0u1 . . . u 3k− 2u0 ,

thenC 0 = u0,1u0,2 . . . u 0,k u1,1u1,2 . . . u 1,k . . . u 3k− 2,1u3k− 2,2 . . . u 3k− 2,k u0,1 ,

where ui,j V ( (u i )), ui V (And (k)), dAnd (k) (u i ) = k ≥ 6, i = 0 , 1, 2, . . . , 3k − 2, andu−

i, 1 , u+i,k V ( (u i )), u+

i,j = ui,j +1 (1 ≤ j ≤ k − 1) and u−i,l = ui,l − 1 (2 ≤ l ≤ k). There

are three cyclic orders

ua,b , u c,d , u e,f , ug,h

V [ (And (k))] according to the denition of the

ring arrangement of the second kind, as follows: ( ua,b , u c,d , u e,f , u g,h ), (ua,b , u e,f , u c,d , ug,h ),(ua,b , u c,d , u g,h , u e,f )(see Fig.1). Let S = {(ua,b , u c,d , u e,f , ug,h ), (ua,b , u e,f , u c,d , ug,h ), (ua,b , u c,d ,ug,h , u e,f )}.

ua,b ua,b ua,b

uc,d uc,due,f ug,h ug,h ue,f

ue,f uc,d ug,h

(ua,b , u c,d , u e,f , ug,h ) (ua,b , u e,f , u c,d , ug,h ) (ua,b , u c,d , u g,h , u e,f )

Fig. 1 Three cyclic orders

Now, we show that 4-ordered hamiltonicity of (And (k)) (k ≥ 6). In fact, we need toprove that α S , there exists a hamiltonian cycle containing α. Without loss of generality,hamiltonian cycle C 0 encounters ( ua,b , u c,d , u e,f , u g,h ) in this order. So we just prove: β S \(ua,b , u c,d , u e,f , u g,h ), there exists a hamiltonian cycle containing β .

According to the Pigeonhole principle, we consider following cases.

Case 1 If these four vertices ua,b , u c,d , u e,f , ug,h are contained in distinct four Cliques of

(And (k)), respectively. And Theorem 2 .1 is true by the result in [5].Case 2 If these four vertices ua,b , u c,d , u e,f , u g,h are contained in a same Clique of (And (k)),then a = c = e = g,b < d < f < h . Let S = {(ua,b , u a,d , u a,f , u a,h ), (ua,b , u a,f , u a,d , u a,h ), (ua,b ,ua,d , u a,h , u a,f )}.

(1) For ( ua,b , u a,d , u a,f , u a,h ) S . C 0 is the hamiltonian cycle that encounters ( ua,b , u a,d , u a,f ,ua,h ) in this order, clearly.




(2) For ( ua,b , u a,f , u a,d , u a,h ) S . We can nd a hamiltonian cycle

C = ua,b C 0 (ua,b , u −a,d )u−

a,d ua,f C 0(ua,f , u a,d )ua,d u+a,f C 0(u+

a,f , u a,h )ua,h C 0(ua,h , u a,b )ua,b

that encounters ( ua,b , u a,f , u a,d , u a,h ) in this order.

(3) For ( ua,b , u a,d , u a,h , u a,f ) S . We can nd a hamiltonian cycle

C = ua, 1C 0(ua, 1 , u−a,f )u−

a,f ua,k C 0(ua,k , u a,f )ua,f (C 1 )(ua,f , u a, 1)ua, 1

that encounters ( ua,b , u a,d , u a,h , u a,f ) in this order by Lemma 2 .2 and Corollary 2 .3 (see Fig.2).

..............C

...................

.............................................

ua, 1

(C 1)

ua,b

ua,d

u−a,f

ua,f ua,h ua,k

C 0

Fig. 2 C = ua, 1C 0(ua, 1 , u−a,f )u−

a,f ua,k C 0(ua,k , u a,f )ua,f (C 1 )(ua,f , u a, 1)ua, 1

Case 3 If these four vertices ua,b , u c,d , u e,f , ug,h are contained in distinct two Cliques of (And (k)), without loss of generality, we assume that ua,b , u c,d V ( (ua )) and ue,f , ug,h

V ( (ue )) in (And (k)) or ua,b , u c,d , u e,f V ( (ua )) and ug,h V ( (ug )) in (And (k)) accord-ing to the notations. Let S = {(ua,b , u a,d , u e,f , u e,h ), (ua,b , u e,f , u a,d , u e,h ), (ua,b , u a,d , u e,h , u e,f )}.

Subcase 3.1 ua,b , u c,d V ( (ua )) and ue,f , ug,h V ( (ue )) in (And (k)).

(1) For ( ua,b , u a,d , u e,f , u e,h ) S . C 0 is the hamiltonian cycle that encounters ( ua,b , u c,d ,ue,f , u g,h ) in this order, clearly.

(2) For ( ua,b , u e,f , u a,d , u e,h ) S . Let C 1 is a hamiltonian cycle in And(k) or And(k) −E (M ), C 2 is a hamiltonian cycle in And(k) −E (C 1 ) or And(k) −E (M ) −E (C 1 )(see Fig.3).

Use A(C 1) to denote a cycle that only through two vertices in (u i )(i = 1 , 2, . . . , 3k −2) andrelated with (C 1 ), and use A(C 2 ) to denote the longest cycle missing the vertex on A(C 1 )in (And (k)) or (And (k)) − M (see Fig.3). We suppose that P 1 = [x, y ], P 2 = [ p, ua,b ] oncycle A(C 1 ) in (And (k)) or (And (k)) − M and P 3 = [m, n ], P 4 = [s, t ] on cycle A(C 2) in

(And (k)) −A(C 1 ) or (And (k)) −M −A(C 1) by Theorem 3 [7] , the analysis of Lemma 2 .4 andthe denition of CEG (see appendix). Now, we have a discussion about the position of vertexx, y, p, s and n in (And (k)).




Fig. 3

In where, C 1 in And (k) or And (k) −E (M ), C 2 in And (k) −E (C 1 ) or And (k) −E (M ) −E (C 1 ),A(C 1 ) in (And (k)) or (And (k))−M , A(C 2) in (And (k))−A(C 1 ) or (And (k))−A(C 1 )−M .

x = ua,b , u a,d ,

y = ue,f ,

p = ue,h , · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·(1)

p = ue,h ,

s, n = ue,h , · · · · · · · · · · · · · · · · · ·· · · · · ·(2)

s = ue,h , . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 3 )

n = ue,h , . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

y = ue,h ,

p = ue,f , · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·(5)

p = ue,f ,

s, n = ue,f , · · · · · · · · · · · · · · · · · ·· · · · · ·(6)

s = ue,f , . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 7 )

n = ue,f , . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)

y = ue,f , u e,h ,

p = ue,f ,

s, n = ue,h , · · · · · · · · · · · · · · · ·· · ·(9)

s = ue,h , . . . . . . . . . . . . . . . . . . . . . (10)

n = ue,h , , . . . . . . . . . . . . . . . . . . . (11)

p = ue,h ,

s, n = ue,f , · · · · · · · · · · · · · · · ·· ·(12)

s = ue,f , . . . . . . . . . . . . . . . . . . . . . (13)

n = ue,f , . . . . . . . . . . . . . . . . . . . . . (14)

p = ue,f , u e,h ,

s = ue,f , · · · · · · · · · · · · · · · ·(15)

n = ue,f , . . . . . . . . . . . . . . . . (16)

s = ue,h , . . . . . . . . . . . . . . . . (17)

n = ue,h , . . . . . . . . . . . . . . . . (18)

s = ue,f , n = ue,h , . . . . . . . (19)

s = ue,h , n = ue,f , . . . . . . . (20)

s = ue,f , n = ue,h , . . . . . . . (21)




For cases (1) and (2), we can nd a hamiltonian cycle

ua,b xP 1 (x, y)ysP 4 (s, t )tG (u a ) [t, m ; ua,b , x]mP 3 (m, n )nG (u e ) [n, p ; y, s ] pP 2( p, ua,b )ua,b

that encounters ( ua,b , u e,f , u a,d , u e,h ) in this order.

For cases (3)-(21), we can nd a hamiltonian cycle that encounters ( ua,b , u e,f , u a,d , u e,h ) inthis order according to the method of (1) and (2).

(3)For cases (2)-(11) and (15)-(21), we can nd a hamiltonian cycle that encounters(ua,b , u a,d , u e,h , u e,f ) in this order according to the method of Case3.1(2).

For case (1), we can nd a hamiltonian cycle

ua,b G (u a ) [ua,b , m ; t]mP ′3(m, n )nG (u e ) [n, p ; y, s ] pysP ′

4(s, t )tu a,b

that encounters ( ua,b , u a,d , u e,h , u e,f ) in this order. P ′i is the path which through the all verticesin (u i )(i = a, . . . , e ) and related with P i (i = 3 , 4) in (And (k)) −A(C 1 ) or (And (k)) −M −A(C 1 )(see Fig.4).

Fig. 4

In where, P 1 , P 2 in (And (k)) or (And (k)) −M , P 3 , P 4 in (And (k)) −A(C 1 ) or (And (k)) −M −A(C 1 ), P ′3 , P ′4 related with P 3 , P 4 in (And (k)) −A(C 1) or (And (k)) −M −A(C 1 ).

For 12-14, we can nd a hamiltonian cycle that encounters ( ua,b , u a,d , u e,h , u e,f ) in thisorder according to the method of 1.

Subcase 3.2 ua,b , u c,d , u e,f V ( (ua )) and ug,h V ( (ug )) in (And (k)). For all condition ,we see the result is proved by the method of Subcase 3 .1.

Case 4 If these four vertices ua,b , u c,d , u e,f , ug,h are contained in distinct three Cliques of

(And (k)). Without loss of generality, we assume that ua,b , u c,d V ( (ua )), ue,f V ( (ue ))and ug,h V ( (ug)) in (And (k)).

(1) For ( ua,b , u a,d , u e,f , u g,h ) S , C 0 is the hamiltonian cycle that encounters ( ua,b , u a,d ,ue,f , u g,h ) in this order, clearly.

(2) For ( ua,b , u e,f , u a,d , ug,h ) S . Let C 1 is a hamiltonian cycle in And(k) or And(k) −E (M ), C 2 is a hamiltonian cycle in And(k) − E (C 1 ) or And(k) − E (M ) − E (C 1), C 3 is a




hamiltonian cycle in And (k) −E (C 1 ) −E (C 2 ) or And (k) −E (M ) −E (C 1 ) −E (C 2)(see Fig.5).Use A(C j ) to denote a cycle that only through two vertices in (u i )(i = 1 , 2, . . . , 3k −2) andrelated with (C j )( j = 1 , 2), and use A(C 3) to denote the longest cycle missing the vertexon A(C 1) and A(C 2 ) in (And (k)) or (And (k)) − M (see Figure5). We can suppose that

P 1 = [uc,d , x], P 2 = [y, u a,b ] on cycle A(C 1 ) in (And (k)) or (And (k)) − M , P 3 = [m, n ],P 4 = [ p, q ] on cycle A(C 2 ) in (And (k)) −A(C 1 ) or (And (k)) −M −A(C 1) and P 5 = [s, t ],

P 6 = [w, z ] on A(C 3 ) in (And (k)) −2

i =1A(C i ) or (And (k)) −M −

2

i=1A(C i ) by Theorem 3 [7] ,

the analysis of Lemma 3 and the denition of CEG (see appendix). Now, we have a discussionabout the position of vertex m, q , x, y, p and n in (And (k)).

m, q = ua,b , u a,d ,

x = ug,h , y = ug,h , · · · · · · · · · · · · · · · · · · · · · · · · · · ·(1)

x = ug,h ,

y = ug,h , · · · · · · · · · · · · · · · · · · · · · · · ·(2)

y = ug,h ;

p = ug,h , · · · · · · · · · · · ·(3)

n = ug,h ,

· · · · · · · · · · · ·(4)

p, n = ug,h . · · · · · · · · ·(5)

Fig. 5

In where, C 1 in And (k) or And (k) −E (M ), C 2 in And (k) −E (C 1 ) or And (k) −E (M ) −E (C 1 ),C 3 in And (k) −E (C 1 ) −E (C 2 ) or And (k) −E (M ) −E (C 1 ) −E (C 2), A(C 1) in (And (k)) or

(And (k)) −M , A(C 2 ) in (And (k)) −A(C 1 ) or (And (k)) −A(C 1 )−M , A(C 3 ) in (And (k)) −A(C 1 ) −A(C 2) or (And (k)) −A(C 1 ) −A(C 2 ) −M .

For case (1), if ue,f V (P i ) (i = 2 , 3, 4), we can nd a hamiltonian cycle that encounters(ua,b , u e,f , u a,d , ug,h ) in this order according to the method of Subcase 3.1,(2).

If ue,f V (P 1), we can nd a hamiltonian cycle

ua,b qP ′4 (q, p) pnP ′3(n, m )mG (u a ) [m, s ; ua,b , q ]sP ′

5(s, t )tG (u g ) [t, y ; p, n, t ]yP ′2(y, u a,b )ua,b or

ua,b mP ′3(m, n )npP ′

4 (q, p)qG (u a ) [q, s; ua,b , m ]sP ′5 (s, t )tG (u g ) [t, y ; p, n, t ]yP ′

2(y, u a,b )ua,b




that encounters ( ua,b , u e,f , u a,d , ug,h ) in this order. There exist some vertices which belong toa same Clique on P 1 , P i and P j (i = 3 , 4; j = 5 , 6). And ue,f V (P ′i )(i = 3 or 4). P ′i isthe path which through the all vertices in (u i )(i = a, . . . , g ) and related with P i (i = 5 , 6) in

(And (k))

−

2

i=1

A(C i ) or (And (k))

−M

−

2

i =1

A(C i ), and missing the vertex on P ′3 , P ′4(refers

to Figure4).

For cases (2)-(5), we can nd a hamiltonian cycle that encounters ( ua,b , u e,f , u a,d , ug,h ) inthis order according to the method of (1).

(3) For cases (1)-(5), we can nd a hamiltonian cycle that encounters ( ua,b , u a,d , ug,h , u e,f )in this order according to the method of Case 4(2).

References

[1] Chris Godsil and Gordon Royle, Algebraic Graph Theory , Springer Verlag, 2004.

[2] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications , North-Holland, NewYork, 1976.

[3] Lenhard N.G., Michelle Schultz, k-Ordered hamiltonian graphs, Journal of Graph Theory ,Vol.24, NO.1, 45-57, 1997.

[4] Faudree R.J., On k-ordered graphs, Journal of Graph Theory , Vol.35, 73-87, 2001

[5] Wang lei,A Yongga, 4-Ordered Hamiltonicity of some Cayley graph , Int.J.Math.Comb. ,l(2007),No.1.117–119.

[6] A Yongga, Siqin, The constrction of Cartasian Product of graph and its perfction, Journal Of Baoji University of Arts and Sciences (Natural Science), 2011(4), 20-23.

[7] Douglas B.West, Introduction to Graph Theory (Second Edition), China Machine Press,2006.

Appendix

By the theorem 1 .9, the analysis of Lemma 2 .4, the denition of CEG, And (k) and the parityof k(s Z + ) , we know that

k = 2 s And (k) (And (k))

s = 1 C 5 C 10

s = 2 And (4) − E (C 1 ) = C 2 (And (4)) − B (C 1 ) = B (C 2 )s = 3 And (6) − E (C 1 ) − E (C 2 ) = C 3 (And (6)) − B (C 1 ) − B (C 2 ) = B (C 3 )

......

...

s = n And (2n ) −n − 1

i =1E (C i ) = C n (And (2n )) −

n − 1

i =1B (C i ) = B (C n )




C 1 C 2

.........................

B (C 1 ) B(C 2 )

s=2

C 1 C 2 C 3

..............

B (C 1 ) B(C 2) B (C 3 )

s=3

If k is odd, it should be illustrated that the M ’s selection method, that is, M satisfycondition ua,b , u c,d , u e,f , u g,h V (M ) in (And (k)). It can be done, because k ≥ 7.

k = 2 s + 1 And (k) (And (k ))

s = 1 And (3) − E (M ) = C 1 (And (3)) − M = B (C 2 1)

s = 2 And (5) − E (M ) − E (C 1 ) = C 2 (And (5)) − E (M ) − B (C 1 ) = B (C 2 )

s = 3 And (7) − E (M ) − E (C 1 ) − E (C 2 ) = C 3 (And (7)) − M − B (C 1 ) − B (C 2 ) = B (C 3 )...

......

s = n And (2n + 1) − E (M ) −

n − 1

i =1 E (C i ) = C n (And (2n + 1)) − M −

n − 1

i =1 B (C i ) = B (C n )

M C 1 M B(C 1 )

s=1

M C 1 B(C 1 )

B (C 2 )

M

s=2




On Equitable Coloring of Weak Product of Odd Cycles

Tayo Charles Adefokun

(Department of Computer and Mathematical Sciences, Crawford University, Nigeria)

Deborah Olayide Ajayi

(Department of Mathematics, University of Ibadan, Ibadan, Nigeria)


Abstract : In this article, we present algorithms for equitable weak product graph of cyclesC m and C n , C m × C n such that it has an equitable chromatic value, χ = (C m × C n ) = 3, with

mn odd and m or n is not a multiple of 3.

Key Words : Equitable coloring, equitable chromatic number, weak product, direct prod-uct, cross product.

AMS(2010) : 05C78

§1. Introduction

Let G be a graph with vertex set V (G) and edge set E (G). A k-coloring on G is a functionf : V (G) → [1, k] = {1, 2, · · · , k}, such that if uv E (G),u ,v V (G) then f (u) = f (v). Avalue χ (G) = k, the chromatic number of G is the smallest positive integer for which G isk-colorable. G is said to be equitably k-colorable if for a proper k-coloring of G with vertexcolor class V 1 , V 2 · · ·V k , then |(|V i | − |V j |)| ≤ 1 for all i, j [i, k ]. Suppose n is the smallestinteger such that G is equitably k-colorable, then n is the equitable chromatic number, χ = (G),of G.

The notion of equitable coloring of a graph was introduced in [6] by Meyer. Notable work onthe subject includes [7] where outer planar graphs were considered and [8] where general planargraphs were investigated. In [1] equitable coloring of the product of trees was considered. Chenet al. in [2] showed that for m, n ≥ 3, χ = (C m ×C n = 2) if mn is even and χ = (C m ×C n = 3)if mn is odd. Recent work include [4], [5]. Furmanczyk in [3] discussed the equitable coloringof product graphs in general, following [2], where the authors separated the proofs of mn into

various parts including the following:

1. m, n odd with n = 0 mod 3

2. m, n odd, with

(a) either m or n , say n satisfying n −1 ≡ 0 mod 31 Received December 8, 2012. Accepted March 16, 2013.



110 Tayo Charles Adefokun and Deborah Olayide Ajayi

(b) either m or n , say n satisfying n −2 ≡ 0 mod 3.

In this paper we present equitable coloring schemes which

1. improve the proof in ( b) above and

2. can be employed in developing the equitable 3-coloring for C m ×C n with mn odd.

§2. Preliminaries

Let G1 and G2 be two graphs with V (G1) and E (G1) as the vertex and edge sets for G1 respec-tively and V (G2) and E (G2) as the vertex and edge sets of G2 respectively. The weak productof G1 and G2 is the graph G1 ×G2 such that V (G1 ×G2) = {(u, v) = u V (G)and u V (G2)}and E (G1 ×G2) =

{(u1v1)(u2v2 ) : u1u2 E (G1)and v1v2 E (G2 )}. A graph P m = u0u1u2 · · ·um − 1 is a path of length m

−1 if for all u i , vj

V (P m ), i

= j. A graph C m = u0u1u2

· · ·um − 1 is a cycle of length

m if for all u i , vj V (C m ), i = j and u0um − 1 E (C m ).The following results due to Chen et al gives the equitable chromatic numbers of product

of cycles.

Theorem 2.1([2]) Let m, n ≥ 3. Then

χ = (C m ×C n ) =2 if mn is even

3 if mn is odd.

We require the following lemma in the main result.

Lemma 2.2 Let n be any odd integer and let n

−1

≡ 0 mod 3. Then n

−1

≡ 0 mod 6.

Proof Since n is odd, then there exists a positive integer m, such that n = 2m + 1. Nowsince n is odd then, n −1 is even. Let 2m ≡ 0 mod 3. Clearly, n ≥ 3. Now 2m = 3k where kis an even positive integer. Thus 2 m = 3(2 k′ ) for some positive integer k ′ and thus 2 m = 6k′ .Hence n −1 = 6 k′ .

§3. Main Results

In this section, we present the algorithms for the equitable 3-coloring of C m ×C n with wherem and n are odd with say n −1 ≡ 0 mod 3 and n −2 ≡ 0 mod 3.

Algorithm 1 Let C m ×C n be product graph and let mn be odd, with n −1 = 0 mod 3.

Step 1 Dene the following coloring for u i vj V (C m ×C n ).

f (u i vj ) =

α 2 for {u i vj : j [n −1]; j ≥ 5; j + 1 = 0 mod 3 }α 1 for {u i vj : j [n −1]; j + 2 = 0 mod 3 } {u i v2 : i [m −1]}α 3 for {u i vj : j [n −1]; j ≥ 6; j = 0 mod 3} {u i v1 , i [m −1]}.



On Equitable Coloring of Weak Product of Odd Cycles 111

Step 2 For all u i v0 ; i [2], dene the following coloring:

(a)

f (u i v0) =α 1 for i = 1

α 2 for i = 0 , 2(b)

f (u i v3) =α 2 for i = 1 , 2

α 3 for i = 0

Step 3 Repeat Step 2(a) and Step 2(b) for all u i v0 and u i v3 for each i [x, x + 2] wherex = 0 mod 3.

Proof of Algorithm 1 Suppose n is odd and n −1 = 0 mod 3. From Lemma 2.2 above,n − 1 = 0 mod 6 and consequently, n − 4 = 0 mod 3. Suppose n − 4

3 = n′ , where n is apositive integer. Let P m ×P n − 4 be a subgraph of P m ×P n , where P n − 4 = v4v5 · · ·vn − 1 .. For

all u i vj V (P m ×P n − 4 ), let

f (u i vj ) =

α 1 for {u i vj : j [n −1], j + 2 = 0 mod 3 }α 2 for {u i vj : j [n −1], j ≥ 5; j + 1 = 0 mod 3 }α 3 for {u i vj : j [n −1]; j ≥ 6; j = 0 mod 3}.

From f (u i vj ) dened above, we see that P m × P n − 4 is equitably 3-colorable with colorset {α 1 , α 2 , α 3} ≡ [1, 3], where |V α 1 | = |V α 2 | = |V α 3 | = mn ′ . Next we show that there existsa 3-coloring of P m ×P 4 that merges with P m ×P n − 4 whose 3-coloring is dened by f (u i vj )above. First, let F (P 3 ×P 4) be the 3- coloring such that

F (P 3 ×P 4) =α 2 α3 α1 α2α 1 α3 α1 α2

α 2 α3 α1 α3

From F (P 3 × P 4) we observe for all j [3], that for F (u0vj ) F (P 3 × P 4), |V α 1 | =1, |V α 2 | = 1 , |V α 3 | = 2; for F (u1vj ) F (P 3 × P 4), |V α 1 | = 2 , |V α 2 | = 1 , |V α 3 | = 1; and forF (u0vj ) F (P 3 ×P 4 ), |V α 1 | = 1 , |V α 2 | = 2 , |V α 3 | = 1 .

We observe, over all, that for F (P 3 ×P 4), |V α 1 | = |V α 2 | = |V α 3 | = 4. These conrm thatP 3 ×P 4 is equitably 3-colorable at every stage of i [2] and that F (P 2 ×P 4) F (P 3 ×P 4) isan equitable 3-coloring of P 2 ×P 4 for both P 2 ×P 4 P 3 ×P 4 . Now the equitable 3-coloring of P m ×P 4 is now obtainable by repeating F (P 3×P 4) at each interval [ x, x +2], where x = 0 mod 3 ,

until we reach m. Clearly, F (u i v3 )∩F (u i v4) = since α1 / F (u i v3). Thus P m ×P n is equitably3-colorable based on the colorings dened earlier. Likewise, F (u i v0) ∩ F (u i un − 1) = sinceα 3 / F (u i v0). Thus P m ×C n is equitably 3-colorable based on the coloring dened above forP m ×P n .

Finally, for any m ≥ 3, the equitable 3-coloring of P m ×P n − 4 with respect to F (P m ×P n − 4)above is equivalent to the equitable 3-coloring of C m ×C n − 4 since u i vj u i vj +1 / E (P m ×P m − 4)for all j [n − 5]. Also, for m ≥ 3 the equitable 3-coloring of P m × P 4 with respect to



112 Tayo Charles Adefokun and Deborah Olayide Ajayi

F (P m ×P 4) above is equivalent to the equitably 3- coloring of C m ×C 4 by mere observation.Thus, C m ×C n is equitably 3- colorable or all positive integer m and odd positive integer nsuch that n −1 = 0 mod 3.

Algorithm 2 Let m or n, say n be odd such that n

−2 = 0 mod 3.

Step 1 Dene the following coloring:

f (u i vj ) =

α1 for {u i vj : j [n −1], j + 1 = 0 mod 3 }α2 for {u i vj : j [n −1], j = 0 mod 3}α3 for {u i vj : j [n −1], j −1 = 0 mod 3}

Step 2(a) For all i [2], let f (u i v0 ) = α1 , α 2 , α 1 respectively α 1 , α 2 [2].

Step 2(b) For all i [2], let f (u i v1) = α3 , α 2 , α 3 respectively, α 3 [2].

Step 3 Repeat step 2(a) and Step 2(b) above for all i [x, x + 2], where x is a positiveinteger and x = 0 mod 3.

Proof of Algorithm 2 Let n be odd and let n−2 = 0 mod 3. By f (u i vj ) in step 1, P m ×P n − 2 ,where P n − 2 = v2v3 · · ·vn − 1 , is equitably 3-colorable with |V α 1 | = |V α 2 | = |V α 3 | = mn ′′ wheren ′′ = n − 2

3 and F (u i v2 ) ∩F (u i vn − 1) = for all i [m −1]. Now, let

F (P 3 ×P 2) =

α 1 α3

α 2 α2

α 1 α3

It is clear that F (P 3 ×P 2) above follows from the coloring dened in step 2 of the algorithmand that F (P 3 ×P 2) is an equitable 3-coloring of P 3 ×P 2 where |V α 1 | = |V α 2 | = |V α 3 | = 2. It isalso clear that F (P 3

×P 2) has an equitable coloring at P 1

×P 2 with

|V α 1

| = 1 ,

|V α 2

| = 0 ,

|V α 3

| = 1

and at P 2 × P 2 with |V α 1 | = 1 , |V α 2 | = 2 , |V α 3 | = 1 . Now, let with x = 0 mod 3. For allx [m − 1], let f (ux vj ) = α1 , α 3 for both j = 0 , 1 respectively; for x + 1 [m − 1], letf (ux +1 vj ) = α2 , for j = 0 , 1 and for x + 2 [m −1], let f (ux +2 vj ) = α1 , α 3 for j = 0 , 1. Withthis last scheme, we have P m ×P 2 that has an equitable 3- coloring for any value of m.

Finally, we can see that P m ×P 2 , for any m, so equitably, 3-colored merges with P m ×P n − 2

that is equitably 3-colored earlier by f (u i vj ), such that F (u i v1 )∩F (u i v2) = for all i [m −1](by a similar argument as in the proof of Algorithm 1) and F (u i v0 ) ∩ F (u i vn − 1) = for alli [m −1] (by a similar argument as in the proof of Algorithm 1).

Likewise C m × C n is equitable 3-colorable (by a similar argument as in the proof of Al-gorithm 1). Therefore, C m

×C n is equitably 3-colorable for any m

≥ 3 and odd n, such that

n −2 = 0 mod 3.

§4. Examples

In Fig.1, we demonstrate how our algorithms equitably color graphs C 5 × C 5 and C 5 × C 7 ,which are two cases that illustrate n −2 = 0 mod 3 and n −1 = 0 mod 3 respectively. In the



On Equitable Coloring of Weak Product of Odd Cycles 113

rst case, we see that χ = (C 5 ×C 5 ) = 3, with |V 1 | = 8 |V 2| = 9 and |V 3| = 8 and in the secondcase, χ = (C 5 ×C 7) = 3, with |V 1| = 12 |V 2| = 11 and |V 3| = 12. (Note that the rst coloringtakes care of the third instance in subcase 2.4 of [2] where it is a special case.)

Fig. 1 Equitable coloring of graphs C 5 ×C 5 and C 5 ×C 7

References

[1] B.-L.Chen, K.-W.Lih, Equitable coloring of Trees, J. Combin. Theory , Ser.B61,(1994)83-37.

[2] B.-L.Chen, K.-W.Lih, J.-H.Yan, Equitable coloring of interval graphs and products of graphs, arXiv:0903.1396v1 .

[3] H.Furmanczyk, Equitable coloring of graph products, Opuscula Mathematica 26(1),2006,

31-44.[4] K.-W.Lih, B.-L.Chen, Equitable colorings of Kronecker products of graphs, Discrete Appl.

Math. , 158(2010) 1816-1826.[5] K.-W.Lih, B.-L.Chen, Equitable colorings of cartesian products of graphs, Discrete Appl.

Math. , 160(2012), 239-247.[6] W. Mayer, Equitable coloring, Amer. Math. Monthly , 80(1973), 920-922.[7] H.P.Yap, Y.Zhang, The equitable coloring of planar graphs, J.Combin. Math. Combin.

Comput. , 27(1998), 97-105.[8] H.P.Yap, Y.Zhang, The equitable ∆-coloring conjecture holds for outer planar graphs,

Bulletin of Inst. of Math. , Academia Sinica, 25(1997), 143-149.




Corrigendum: On Set-Semigraceful Graphs

Ullas Thomas

Department of Basic Sciences, Amal Jyothi College of Engineering

Koovappally P.O.-686 518, Kottayam, Kerala, India

Sunil C Mathew

Department of Mathematics, St.Thomas College Palai

Arunapuram P.O.-686 574, Kottayam, Kerala, India


In this short communication we rectify certain errors which are in the paper, On Set-SemigracefulGraphs, International J. Math. Combin. , Vol.2(2012), 59-70. The following are the correct ver-sions of the respective results.

Remark 3.2 (5) The Double Stars ST (m, n ) where |V | is not a power of 2, are set-semigracefulby Theorem 2 .13.

Remark 3.5 (3) The Double Stars ST (m, n ) where m is odd and m + n + 2 = 2 l , are notset-semigraceful by Theorem 2 .12.Delete the following sentence below Remark 3.9: ”In fact the result given by Theorem 3 .3holds for any set-semigraceful graph as we see in the following”.

Theorem 4.8([3]) Every graph can be embedded as an induced subgraph of a connected set-graceful graph.

Since every set-graceful graph is set-semigraceful, from the above theorem it follows that

Theorem 4.8A Every graph can be embedded as an induced subgraph of a connected set-semigraceful graph.

However, below we prove:

Theorem 4.8B Every graph can be embedded as an induced subgraph of a connected set-semigraceful graph which is not set-graceful.

Proof Any graph H with o(H ) ≤ 5 and s(H ) ≤ 2 and the graphs P 4 , P 4 K 1 , P 3 K 2 andP 5 are induced subgraphs of the set-semigraceful cycle C 10 which is not set-graceful. Again any

1 Received January 8, 2013. Accepted March 22, 2013.



Corrigendum: On Set-Semigraceful Graphs 115

graph H ′ with 3 ≤ o(H ′ ) ≤ 5 and 3 ≤ s(H ′ ) ≤ 9 can be obtained as an induced subgraph of H 1 K 1 for some graph H 1 with o(H 1) = 5 and 3 ≤ s(H 1) ≤ 9. Then 3 < log2(|E (H 1 K 1)|+1) < 4, since 8 ≤ s(H 1 K 1) < 15 and hence H 1 K 1 is not set-graceful. By Theorem 2 .4,

4 =

log

2(

|E (H 1

K 1)

|+ 1)

≤ γ (H 1

K 1)

≤ γ (K 6) (by Theorem 2 .5)

= 4 (by Theorem 2 .19)

So that H 1 K 1 is set-semigraceful. Further, note that K 5 is set-semigraceful but not set-graceful.

Now let G = ( V, E ); V = {v1 , . . . , vn } be a graph of order n ≥ 6. Consider a set-indexer g of G with indexing set X = {x1 , . . . , x n } dened by g(vi ) = {x i}; 1 ≤ i ≤ n. LetS = {g(e) : e E } {g(v) : v V }. Note that |S | = |E |+ n. Now take a new vertex u and joinwith all the vertices of G. Let m be any integer such that 2 n − 1 < m < 2n −(|E |+ n +1). Since

|E

| ≤

n(n −1)2

and n

≥ 6, such an integer always exists. Take m new vertices u1 , . . . , u m and

join all of them with u. A set-indexer f of the resulting graph G ′ can be dened as follows:

f (u) = , f (vi ) = g(vi ); 1 ≤ i ≤ n.

Besides, f assigns the vertices u1 , . . . , u m with any m distinct elements of 2 X \ (S ).Thus, γ (G ′ ) ≤ n. But we have 2n > |E | + n + m + 1 > m > 2n − 1 so that γ (G ′ ) ≥ n, byTheorem 2 .4. Hence,

log2(|E (G ′ )|+ 1) < log2(|E (G ′ )|+ 1) = n = γ (G ′ ).

This shows that G ′ is set-semigraceful, but not set-graceful.

Corollary 4.16 The double fan P k K 2 where k = 2n

− m and 2n

≥ 3m; n ≥ 3 is set-semigraceful.

Proof Let G = P k K 2 ; K 2 = ( u1 , u2). By Theorem 2 .4, γ (G) ≥ log2(|E |+ 1) =log2(3(2n −m) + 1) = n + 2. But, 3 m ≤ 2n m < 2n − 1 −1. Therefore,

2n −(2n − 1 −2)) ≤ 2n −m < 2n −1

2n − 1 + 1 ≤ 2n −m −1 < 2n −2

2n − 1 + 1 ≤ k −1 < 2n −2; k = 2 n −m

2n − 1 + 1 ≤ |E (P k )| < 2n

log2(

|E (P k )

|+ 1) = n

γ (P k ) = n

since P k is set-semigraceful by Remark 3 .2(3).

Let f be a set-indexer of P k with indexing set X = {x1 , . . . , x n }. Dene a set-indexer g of G with indexing set Y = X {xn +1 , xn +2 } as follows:

g(v) = f (v) for every v V (P k ), g(u1) = {xn +1 } and g(u2) = {xn +2 }.



116 Ullas Thomas and Sunil C Mathew

Corollary 4.17 The graph K 1,2n − 1 K 2 is set-semigraceful.

Proof The proof follows from Theorems 4 .15 and 2.33.

Theorem 4.18 Let C k where k = 2 n

−m and 2n + 1 > 3m; n

≥ 2 be set-semigraceful. Then

the graph C k K 2 is set-semigraceful.

Proof Let G = C k K 2 ; K 2 = ( u1 , u2). By theorem 2.4, γ (G) ≥ log2(|E |+ 1) =log2(3(2n −m) + 2) = n + 2. But, 3 m ≤ 2n + 1 m < 2n − 1 . Therefore,

2n −(2n − 1 −1)) ≤ 2n −m < 2n

2n − 1 + 1 ≤ k < 2n ; k = 2 n −m

2n − 1 + 1 ≤ |E (C k )| < 2n

log2(|E (C k )|+ 1) = n

γ (C k ) = n

since C k is set-semigraceful.

Let f be a set-indexer of C k with indexing set X = {x1 , . . . , x n }. Dene a set-indexer g of G with indexing set Y = X {xn +1 , xn +2 } as follows:

g(v) = f (v) for every v V (C k ), g(u1) = {xn +1 } and g(u2) = {xn +2 }.

Corollary 4.21 W n where 2m −1 ≤ n ≤ 2m + 2 m − 1 −2; m ≥ 3 is set-semigraceful.

Proof The proof follows from Theorem 3 .15 and Corollary 4 .20.

Theorem 4.22 If W 2k where 2n − 1

3 ≤ k < 2n − 2; n ≥ 4 is set-semigraceful, then the gear graph

of order 2k + 1 is set-semigraceful.

Proof Let G be the gear graph of order 2 k + 1. Then by theorem 2.4,

log2(3k + 1) ≤ γ (G) ≤ γ (W 2k ) (by Theorem 2 .5)

= log2(4k + 1) (since W2k is set −semigraceful)

= log2(3k + 1)

since

2n − 1

3 ≤ k < 2n − 2 2n − 1 ≤ 3k < 4k < 2n

2n − 1

+ 1 ≤ 3k + 1 < 4k + 1 ≤ 2n

.

Thusγ (G) = log2 (|E |+ 1) .

So that G is set-semigraceful.





First International Conference

On Smarandache Multispace and Multistructure

Organized by Dr.Linfan Mao, Academy of Mathematics and Systems, Chinese Academy of Sciences, Beijing 100190, P.R.China. In American Mathematical Society’s Calendar website:

http://www.ams.org/meetings/calendar/2013 jun28-30 beijing100190.html

June 28-30, 2013, Send papers by June 1, 2013 to Dr.Linfan Mao by regular mail to the abovepostal address, or by email to [email protected].

A Smarandache multispace (or S-multispace ) with its multistructure is a nite or innite(countable or uncountable) union of many spaces that have various structures. The spacesmay overlap, which were introduced by Smarandache in 1969 under his idea of hybrid science:combining different elds into a unifying eld , which is closer to our real life world since we livein a heterogeneous space. Today, this idea is widely accepted by the world of sciences.

The S-multispace is a qualitative notion, since it is too large and includes both metricand non-metric spaces. It is believed that the smarandache multispace with its multistructureis the best candidate for 21st century Theory of Everything in any domain. It unies manyknowledge elds. A such multispace can be used for example in physics for the Unied Field Theory that tries to unite the gravitational, electromagnetic, weak and strong interactions.Or in the parallel quantum computing and in the mu-bit theory, in multi-entangled statesor particles and up to multi-entangles objects. We also mention: the algebraic multispaces(multi-groups, multi-rings, multi-vector spaces, multi-operation systems and multi-manifolds,geometric multispaces (combinations of Euclidean and non-Euclidean geometries into one spaceas in Smarandache geometries), theoretical physics, including the relativity theory, the M-theoryand the cosmology, then multi-space models for p-branes and cosmology, etc.

The multispace and multistructure were rst used in the Smarandache geometries (1969),which are combinations of different geometric spaces such that at least one geometric axiombehaves differently in each such space. In paradoxism (1980), which is a vanguard in literature,arts, and science, based on nding common things to opposite ideas, i.e. combination of con-tradictory elds. In neutrosophy (1995), which is a generalization of dialectics in philosophy,and takes into consideration not only an entity < A > and its opposite < Anti A > as dialec-tics does, but also the neutralities ¡neutA¿ in between. Neutrosophy combines all these three< A > , < Anti A > , and < neut A > together. Neutrosophy is a metaphilosophy, includingneutrosophic logic , neutrosophic set and neutrosophic probability (1995), which have, behind theclassical values of truth and falsehood, a third component called indeterminacy (or neutrality,

which is neither true nor false, or is both true and false simultaneously - again a combinationof opposites: true and false in indeterminacy). Also used in Smarandache algebraic structures(1998), where some algebraic structures are included in other algebraic structures.

All reviewed papers submitted to this conference will appear in its Proceedings , publishedin USA this year.



March 2013

Contents

Global Stability of Non-Solvable Ordinary Differential Equations

With Applications BY LINFAN MAO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 01

m th -Root Randers Change of a Finsler Metric

BY V.K.CHAUBEY AND T.N.PANDEY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Quarter-Symmetric Metric Connection On Pseudosymmetric Lorentzian

α-Sasakian Manifolds BY C.PATRA AND A.BHATTACHARYYA . . . . . . . . . . . . . . . 46

The Skew Energy of Cayley Digraphs of Cyclic Groups and Dihedral

Groups BY C.ADIGA, S.N.FATHIMA AND HAIDAR ARIAMANESH . . . . . . . . . . . . . 60Equivalence of Kropina and Projective Change of Finsler Metric

BY H.S.SHUKLA, O.P.PANDEY AND B.N.PRASAD . . . . . . . . . . . . .. .. . . . . . . . . . . . . . . . . 77

Geometric Mean Labeling Of Graphs Obtained from Some Graph

Operations BY A.DURAI BASKAR, S.AROCKIARAJ AND B.RAJENDRAN. . . . . 85

4-Ordered Hamiltonicity of the Complete Expansion Graphs of

Cayley Graphs BY LIAN YING, A YONGGA, FANG XIANG AND SARULA . . . . 99

On Equitable Coloring of Weak Product of Odd Cycles

BY TAYO CHARLES ADEFOKUN AND DEBORAH OLAYIDE AJAYI. . . . . . . . . . . . 109

Corrigendum: On Set-Semigraceful Graphs BY ULLAS THOMAS AND SUNIL C MATHEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Documents

MATHEMATICAL COMBINATORICS (INTERNATIONAL BOOK SERIES), Volume 1 / 2013