WIENER INDICES OF BENZENOID GRAPHS* · chemistry [1]. It was introduced in 1947 by Harold Wiener, then a chemistry student at the Brooklyn College, as the path number for characterization

Bulletin of the Chemists and Technologists of Macedonia, Vol. 23, No. 2, pp. 113–129 (2004) GHTMDD – 444 ISSN 0350 – 0136 Received: February 20, 2004 UDC: 547 : 541 Accepted: March 18, 2004

Original scientific paper

WIENER INDICES OF BENZENOID GRAPHS*

Damir Vukičević1, Nenad Trinajstić2

1Department of Mathematics, University of Split, Teslina 12, HR-21000 Split, Croatia

2The Rugjer Bošković Institute, P. O. Box 180, HR-10001 Zagreb, Croatia

[email protected]

Formulae are derived for Wiener indices of a class of pericondensed benzenoid graphs consisting of three rows of hexagons of various lengths. A corresponding computer program is prepared. Results obtained are checked against numbers computed by Pascal-oriented pseudocode that is based on ring-matrices of benzenoid graphs. This program can compute the Wiener index of any benzenoid system and not just for pericondensed benzenoids considered in this paper.

Keywords: benzenoid graph; pericondensed benzenoids; ring-matrix; Wiener index

INTRODUCTION

The Wiener index W is the first topological index (graph-theoretical invariant) to be used in chemistry [1]. It was introduced in 1947 by Harold Wiener, then a chemistry student at the Brooklyn College, as the path number for characterization of alkanes [2,3]. In a chemical language, the Wiener index is equal to the sum of all shortest carbon-carbon bond paths in a molecule. In a graph-theoretical language, the Wiener index is equal to the count of all shortest distances in a (molecular) graph. First mathematical definition of this index, based on the concept of graph-theoretical distance as encoded in the distance matrix [4] is due to Ho-soya [5]. Since its inception the Wiener index was used in a numerous structure-property studies [6–10].

Explicit formulas for several classes of ben-zenoid graphs were also proposed in recent years [11–18]. Note that benzenoid graphs are graph-theoretical representions of benzenoid hydrocar-

bons [19, 20]. In this paper we give formulas for the calculation of the Wiener index of pericon-densed benzenoid graphs made up from three rows of hexagons of various lengths (in a chemical lan-guage a subclass of pericondensed benzenoids) [19, 21]. A pericondensed benzenoid graph is a benzenoid graph in which internal vertices appear, that is, vertices that belong to three hexagons. Coronene and a number of its derivatives, such as dibenzo[a,j]coronene, dibenzo [bc,kl]coronene, naphtho[2,3-a]coronene, enumerous derivatives of perylene, anthathrene, peropyrene, etc., such as benzo[ghi]perylene, dibenzo[fg,op]anthathrene, naph-tho[2,1,8-bcd]peropyrene and many other kinds of benzenoids belong, for example, to the studied class of pericondensed benzenoids. They possess interesting mathematical, physical, chemical, bio-logical and technological properties [22–25].

In the application section we give a Pascal-oriented pseudocode by which the Wiener index for any benzenoid graph can be calculated. *Dedicated to the 30th anniversary of this journal.

114 D. Vukičević, N. Trinajstić

Bull. Chem. Technol. Macedonia, 23, 2, 113–129 (2004)

DERIVATION OF FORMULAS

Let , , .a b c N∈ Denote by ( ), ,G a b c a pericondensed benzenoid graph given below:

c benzenoid rings

b benzenoid rings a benzenoid rings

First, we need to prove several Lemmas that will imply our main result. Lemma 1: Let n N∈ and let ( )L n be a benzenoid chain depicted below:

n - times

Its Wiener index is given by 3

226 16( ( )) 1 12 .3 3n nW L n n= + + +

Proof: Label vertices of ( )L n as on the following benzenoid chain:

x 1 x 2 x 3

x 2n+1

y 1 y 2 y 3 y 2n+1

...

...

Also denote

{ }{ }

1 2 2 1

1 2 2 1

, ,...,

, ,..., .n

n

X x x x

Y y y y+

+

=

= Then

{ } { }, ,

2 2 1 2 2 1 2 1 2 2 1

1 1 1 1 1 1 1 1

32

( ( )) ( , ) ( , ) ( , )

( ) ( ) ( 1) ( 1) 2

26 161 12 .3 3

u v X u v Y u Xv Y

n n n n n i n n

i j i i j i i j i j i

W L n d u v d u v d u v

j i j i i j j i n

n nn

⊆ ⊆ ∈∈

+ + + +

= = + = = + = = = = +

= + +

= − + − + − + + − + +

= + + +

∑ ∑ ∑

∑∑ ∑∑ ∑∑ ∑∑

□ Lemma 2: Let n N∈ . Then

32896 64( ( 1, , 1)) 198 136 .

3 3n nW G n n n n+ + = + + +

Wiener indices of benzenoid graphs 115

Glas. hem. tehnol. Makedonija, 23, 2, 113‡129 (2004)

Proof: Label vertices of ( 1, , 1)G n n n+ + as on the following pericondensed benzenoid graph:

x 1,0 x 1,1 x 1,2 x 1,2n+2

x 2,0 x 2,1 x 2,2 x 2,2n+2

x 3,2n+2

x 4,2n+2

x 3,0 x 3,1 x 3,2

x 4,0

x 4,1

x 4,2

...

Also denote

{ }{ }{ }{ }

1 1,0 1,1 1,2 1,2 2

2 2,0 2,1 2,2 2,2 2

3 3,0 3,1 3,2 3,2 2

4 4,0 4,1 4,2 4,2 2

, , ,...,

, , ,...,

, , ,...,

, , ,..., .

n

n

n

n

S x x x x

S x x x x

S x x x x

S x x x x

+

+

+

+

=

=

=

=

Then

{ }{ }

( ) ( ) ( )( )( )

1 2 3 4 1 23 4

1 1 24 3 3

, ,

2 2 1

0 0

( ( 1, , 1))( , ) ( , ) ( , )

( ( 1)) ( ( 1)) ( , ) ( , ) 2 ( , )

2 ( ( 1)) 3 6( 2) 2 2 4 2 2 2 2 1 2 2

u v S S u v S S u S Sv S S

u S u S u Sv S v S v S

n n

i j

W G n n nd u v d u v d u v

W L n W L n d u v d u v d u v

W L n i j n n

⊂ ∪ ⊂ ∪ ∈ ∪∈ ∪

∈ ∈ ∈∈ ∈ ∈

+ +

= =

+ + =

= + +

= + + + + + +

= + + − + + + − − + + + + + + − −

∑ ∑ ∑

∑ ∑ ∑

∑

( ) ( ) ( ) ( )

1

2 1 2 1 2 1 2 1

0 0 0 0

2 2 8 1 2 1 2 2 .n n n n

i j i j

i j n i j n+ + + +

= = = =

+

+ ⋅ − + + + + + − + + +

∑

∑∑ ∑∑

Using the Lemma 1 and eliminating the signs of the absolute value, we get

( )( )( ) ( ) ( )

( ) ( )( )( )( )

( ) ( )( )( )

( ) ( ) ( )( )( )( )

( )

32

2 1 2 1 1 2 1

0 0 0 1

2 1 2 1 1 2 1

0 0 0 1

0

1, , 1

26 1 16 12 1 12 1

3 3

3 3 6( 2) 2 2 4 2 2 2 2 1 2 2

2 2 2 8 1 2

1

n b bi

i j i j i

n b bi

i j i j i

i

j

W G n n n

n nn

i j j i n n

i j j i n

i j

+ + − +

= = = = +

+ + − +

= = = = +

=

+ + =

+ + = ⋅ + + + + +

+ − + + − + + + − − + + + + + + − − +

+ ⋅ − + + − + + + + +

+ − + +

∑ ∑ ∑ ∑

∑ ∑ ∑ ∑

∑ ( )( )( )( )

( )2 1 2 1 1 2 1

0 0 1

1 2 2 .n b b

i i j i

j i n+ + − +

= = = +

− + + +

∑ ∑ ∑



After some involved algebraic manipulations, we get

32896 64( ( 1, , 1)) 198 136 .

3 3n nW G n n n n+ + = + + +

□ Lemma 3: Let , ,a b c N∈ such that .b c a< ≤ Then we have

( )( )3 3

2 2 2

32 2 2 2

110 16 52 110, , 26 20 16 16 32 643 3 3 3 3

1616 16 16 64 32 20 16 .3

a a b b cW G a b c a ab b ab

cac a c bc abc b c c ac

= + + + + + − + − + +

+ + − + + + +

Proof: Label vertices of ( , , )G a b c as on the following pericondensed benzenoid graph:

x 1,1

x 1,2x 1,3x 1,4x 2,1

x 2,2x 2,3x 2,4

x 3,0x 3,1

x 3,2(c – b - 1) - 1 ...

x 4,0

x 4,1... x 4,2(c - b - 1) - 1 x 5,0

x 5,1

x 5,2 x 5,2(a – b - 1) - 1...

x 6,1x 6,2x 6,0 ... x 6,2(a – b - 1) - 1

x 7,0x 8,0

x 9,0x 10,0

x 10,1

x 9,1

x 8,1

x 7,1

...

... x 7,2b x 8,2b

x 9,2b x 10,2b

Also denote

{ }{ }

( ){ }( ){ }( ){ }( ){ }

{ }{ }

1 1,1 1,2 1.3 1,4

2 2,1 2,2 2,3 2,4

3 3,0 3,1 3,2 1 1

4 4,0 4,1 4,2 1 1

5 5,0 5,1 5,2 1 1

6 6,0 6,1 6,2 1 1

7 7,0 7,1 7,2

8 8,0 8,1 8,2

, , ,

, , ,

, ,...,

, ,...,

, ,...,

, ,...,

, ,...,

, ,...,

c b

c b

a b

a b

b

b

S x x x x

S x x x x

S x x x

S x x x

S x x x

S x x x

S x x x

S x x x

− − −

− − −

− − −

− − −

=

=

=

=

=

=

=

=

{ }{ }

9 9,0 9,1 9,2

10 10,0 10,1 10,2

, ,...,

, ,..., .b

b

S x x x

S x x x

=

= Then



( )( )

( ){ }

( ){ }

( ){ }

( ){ }

( ){ }

( ) ( )

( ) ( )

1 2 7 8 9 10 7 8 1 3 4 7 8 1

9 10 2 5 6 9 10 2 3 4 3 45 6 2

15 6

, , ,

, ,

, ,

, , ,

, , , ,

, ,

u v S S S S S S u v S S S S S u v S S S

u v S S S S S u v S S S u S S u S Sv S S v S

u S uv S S

W G a b c

d u v d u v d u v

d u v d u v d u v d u v

d u v d u v

⊂ ∪ ∪ ∪ ∪ ∪ ⊂ ∪ ∪ ∪ ∪ ⊂ ∪ ∪

⊂ ∪ ∪ ∪ ∪ ⊂ ∪ ∪ ∈ ∪ ∈ ∪∈ ∪ ∈

∈∈ ∪

=

= + − +

+ − + + +

+ +

∑ ∑ ∑

∑ ∑ ∑ ∑

∑ ( )

( )( ) ( )( ) ( )( )( ) ( )( ) ( )( )( )

( )( )( )

( )( )

( )( )

( )

3 4 5 67 8 9 10 7 8 9 10

2 1 1 2 1 1 2 1 1

0 0 0

2 1 1 2

0 0

,

1, , 1

4 4 5 6 6 7 3 4 5 6 4 5 6 7 8

3 4 5 6 4 5 6 7 8 4 4 4 5 5 6

S S u S Sv S S S S v S S S S

c b a b c b

i j i

a b c b

i j

d u v

W G b b b W L a W L b W L c W L b

i j i

i i j

∈ ∪ ∈ ∪∈ ∪ ∪ ∪ ∈ ∪ ∪ ∪

− − − − − − − − −

= = =

− − − − −

= =

+

= + + + − + − +

+ + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + +

∑ ∑

∑ ∑ ∑

∑( )

( )( )

1 12

0

2 1 12

0 0

4 4 4 5 5 6 .

b

i

a bb

i j

i j

−

=

− − −

= =

+

+ + + + + +

∑ ∑

∑ ∑

Using Lemmas 1 and 2, we get

( )( )

( )( )( )

3 3 32 2 2

3 32 2

2 1 1 2 1 1

0 0

, ,

896 64 26 16 26 16198 136 1 12 1 123 3 3 3 3 3

26 16 26 161 12 1 123 3 3 3

4 4 5 6 6 7 3 4 5 6 4 5 6 7c b a b

i j

W G a b c

b b a a b bb a b

c c b bc b

i j− − − − − −

= =

=

+ + + + + + + − + + + +

+ + + + − + + + +

+ + + + + + + + + + + + + + +∑ ∑ ( )( )

( )( )

( )( )

( )( )

2 1 1

0

2 1 1 2 1 12

0 0 0

2 1 12

0 0

8

3 4 5 6 4 5 6 7 8 4 4 4 5 5 6

4 4 4 5 5 6 .

c b

i

a b c bb

i i j

a bb

i j

i

i i j

i j

− − −

=

− − − − − −

= = =

− − −

= =

+

+ + + + + + + + + + + + + + + +

+ + + + + +

∑

∑ ∑ ∑

∑ ∑

After a straightforward algebra, we get

( )( )3 3

2 2 2

32 2 2 2

110 16 52 110, , 26 20 16 16 32 643 3 3 3 3

1616 16 16 64 32 20 16 .3

a a b b cW G a b c a ab b ab


= + + + + + − + − + +

+ + − + + + +

□



Lemma 4: Let , ,a b c N∈ such that .c b a≤ < Then

( )

.3

2848648820

3134

32081884

3100

31624

39830),,(

322222

3222

32

cbcacccbbccaac

cbabbbaabbaaacbaGW

+−−+++++

+++−+++++++=

Proof: Note that ( ), ,G a b c is a subgraph of pericondensed benzenoid graph ( ), , 1G a b b + . Label ver-

tices of graphs ( , , )G a b c and ( ), , 1G a b b + as on the following graph:

x 1,0 x 2,0

x 3,0 x 4,0 x 4,1

x 3,1

x 2,1

x 1,1 x 1,2 x 2,2

x 3,2 x 4,2

...

...

x 4,2c

x 3,2c

x 2,2c

x 1,2c

x 5,0x 5,1

x 5,2x 5,t

x 6,t

...

x 6,0

x 6,1x 6,2x 7,0

x 7,1

x 7,2x 7,t

x 8,0

x 8,1

x 8,2

x 8,t

... x 10,0

x 10,1x 10,2

x 9,0 x 9,2x 9,2a - 2b - 3x 9,1x 10,2a - 2b - 3

where 2( 1 ) 1t b c= + − − . Also denote

{ }{ }{ }{ }

( ){ }( ){ }

( ){ }( ){ }

1 1,0 1,1 1,2

2 2,0 2,1 2,2

3 3,0 3,1 3,2

4 4,0 4,1 4,2

5 5,0 5,1 5,2 1 1

6,1 6,0 6,1 6,2 1 1 1

6,2 6,2 1 1

7 7,0 7,1 7,2 1 1

8 8,0

, ,...,

, ,...,

, ,...,

, ,...,

, ,...,

, ,...,

, ,...,

,

c

c

c

c

b c

b c

b c

b c

S x x x

S x x x

S x x x

S x x x

S x x x

S x x x

S x

S x x x

S x

+ − −

+ − − −

+ − −

+ − −

=

=

=

=

=

=

=

=

= ( ){ }( ){ }

( ){ }

8,1 8,2 1 1

9 9,0 9,1 9,2 1 1

10 10,0 10,1 10,2 1 1

,...,

, ,...,

, ,..., .

b c

a b

a b

x x

S x x x

S x x x

+ − −

− − −

− − −

=

=

Then



( )( )( )

{ }( ) ( )

( ) ( ) ( ) ( ){ }

( ) ( )

( )

1 2 3 4 5 6,1 6,2 7 8 9 10 5 51 2 3 4 6,1 6,2

5 5 5 5 6,2 6,27 8 9 10 1 6,1 2

,

,

, ,

, , ,

, , , , , ,

,

u v S S S S S S S S S S S u S u Sv S S S S v S S

u S u S u S u v S u S u Sv S v S v S S v S v S S

u

W G a b c

d u v d u v d u v

d u v d u v d u v d u v d u v d u v

d u v

⊂ ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∈ ∈∈ ∪ ∪ ∪ ∈ ∪

∈ ∈ ∈ ∈ ∈ ∈∈ ∈ ∈ ∪ ∈ ∈ ∪

=

= −

− − − − − − −

−

∑ ∑ ∑

∑ ∑ ∑ ∑ ∑ ∑

( )

( )( ) ( )( )

( ) ( )( )( )

( ) ( )( )( )

( ) ( )

6,2 6,23 7 9 4 8 10

2 1 12

0 0

2 1 1 2 1 1 2 1 1 2 1 1

0 0 0 0

,

, , 1 1 2 3 4 4 4 2 4

1 2 1 2 8 1 2

3 6 1 2 4 2

S u Sv S S S v S S S

b cc

i j

b c b c b c b c

i j i j

d u v

W G a b b i j

i j b c i j b c

i j b c

∈ ∈∈ ∪ ∪ ∈ ∪ ∪

+ − −

= =

+ − − + − − + − − + − −

= = = =

−

= + − + + + + + + + −

− − + + + − − − + + + − − −

− − + + + − − + +

∑ ∑

∑ ∑

∑ ∑ ∑ ∑

( )( )( )( )

( ) ( )( )( )( )

( ) ( )( ) ( )( )

( ) ( )( )

( )

2 1 1 2 1 1

0 0

2 1 1 2 1 1 2 1 1

2 0 0

2 1 1 2 2 1

0 0 0 0

2 12

0 0

2 2 2 1 6

3 4 2 2 5 6 6 7 4

1 2 1 1

3 2 3 4 5 4

b c b c

i j

b c a b a b

i j i

b c i c b

i j i i

a bb

i i

b c

i j i

i j b c i i

i i i

+ − − + − −

= =

+ − − − − − − − −

= = =

+ − − +

= = = =

− −

= =

+ + + + − − −

− + + + + + + + + −

− − − + + − + − + −

− + + + + − + + + +

∑ ∑

∑ ∑ ∑

∑ ∑ ∑ ∑

∑ ∑ ( )( )2 12

0 0

.a bb

i i

i− −

= =

∑ ∑

Using the last Lemma and eliminating the signs of absolute value, we get

( )( )

( )( ) ( )( )

( ) ( )( )( )( )

( )

( ) ( )( )( )( )

( )

2 1 12

0 0

2 1 1 2 1 1 1 2 1 1

0 0 0 1

2 1 1 2 1 1 1 2 1 1

0 0 0 1

, ,

, , 1 1 2 3 4 4 4 2 4

1 1 2 1

2 2 8 1 2

b cc

i j

b c b c b ci

i j i j i

b c b c b ci

i j i j i

W G a b c

W G a b b i j

i j j i b c

i j j i b c

+ − −

= =

+ − − + − − − + − −

= = = = +

+ − − + − − − + − −

= = = = +

=

= + − + + + + + + + −

− − + + − + + + − −

− − + + − + + + − −

∑ ∑

∑ ∑ ∑ ∑

∑ ∑ ∑ ∑

( ) ( )( )( )( )

( ) ( )

( ) ( )( )( )( )

( ) ( )( ) ( )( )

2 1 1 2 1 1 1 2 1 1

0 0 0 1

2 1 1 2 1 1 2 1 1

2 0 0

2 1 2 2 1

0 0 0 0

3 3 6 1 2 12 1 6

3 4 2 2 5 6 6 7 4

1 2 1 1

b c b c b ci

i j i j i

b c a b a b

i j i

b c i c b

i j i i

i j j i b c b c

i j i

i j b c i i

+ − − + − − − + − −

= = = = +

+ − − − − − − − −

= = =

+ − − +

= = = =

−

− − + + − + + + − − + + − − −

− + + + + + + + + −

− − − + + − + − +

∑ ∑ ∑ ∑

∑ ∑ ∑

∑ ∑ ∑

( ) ( )( )

( ) ( )( )

1

2 1 2 12 2

0 0 0 03 2 3 4 5 4 .

a b a bb b

i i i ii i i i

− − − −

= = = =

−

− + + + + − + + + +

∑

∑ ∑ ∑ ∑



After a rather involved algebra, we get

( )( )3 3

2 2 2 2

32 2 2 2 2

98 16 100 20b 134c, , 30 24 4 8 18 8 + + +3 3 3 3 3

28c+20ac +8a 8 4 c+6c -8ac -4bc + . 3

a a bW G a b c a ab a b b ab

c bc b

= + + + + + + + −

+ +

□ Lemma 5: Let ,a b N∈ such that .b a≥ Then

( )( )3

2 3 2 2 2146 16, , 31 82 20 16 40 16 28 16 .3 3

b bW G a b a a a a ab a b b ab= + + + + + − + + +

Proof: Denote by 1S the set of vertices marked by black dots on the following pericondensed benze-

noid graph and denote by 2S the set of remaining vertices:

Then

( )( ) ( ){ }

( ) ( ){ }

( )( ) ( )( )

( )( )

2 1 12

, ,

22

0 0

, , , , ,

, 1, 1 2 2 3 1 2 2 3 8 8 .

u v S u S u v Su S

b aa

i j

W G a b a d u v d u v d u v

W G a a a i j W L b a

⊂ ∈ ⊂∈

−

= =

= + +

= − + + + + + + + + + + + −

∑ ∑ ∑

∑ ∑


( )( ) ( ) ( ) ( )

( )( )

( ) ( )

32

3222

0 0

896 1 64 1, , 198 136 1

3 3

1626( )1 2 2 3 1 2 2 3 8 8 1 12 .3 3

b aa

i j

a aW G a b c a

b ab ai j b a−

= =

− − = + + − + +

−−+ + + + + + + + + + + + + − +∑ ∑

After involved, but straightforward algebraic manipulations, we get

( )( )3

2 3 2 2 2146 16, , 31 82 20 16 40 16 28 16 .3 3

b bW G a b a a a a ab a b b ab= + + + + + − + + +

□ Lemma 6: Let , ,a b c N∈ such that .c a b≤ ≤ Then



( )( )3 3

2 2 2 2

32 2 2 2 2

97 20 146 16 149, , 35 4 20 8 28 83 3 3 3 3

2812 4 20 8 4 4 8 .3

a a b b cW G a b c a ab a b b ab

cac a c bc b c c ac bc

= + + + + + − + + + + +

+ + + + + − − +

Proof: Denote by 1S the set of vertices marked by black dots on the following pericondensed benze-

noid graph, denote by 2S the single vertex marked by black square, denote by 3S the set of vertices marked

by black triangles and by 4S the set of remaining vertices:

Then

( )( ) ( ){ }

( ) ( ) ( ){ }

( ) ( )

( )( ) ( )( )

( ) ( )( )( )

( )( )

( )

3 4 1 1 1 2 22 4 3 4 3

, ,

2 2

0 0

2 2

0 0

2

0

, , , , , , , ,

, 1, 1 2 3 1 2 2 6 6

2 2 2 3 2 2

1 2 3 4 5 2 3 4 3

u v S S u S u S u v S u S u Sv S S v S v S v S

b a a

i j

b a c

i j

i

W G a b c d u v d u v d u v d u v d u v d u v

W G a a c i j

a c a c i j W L b a

i

⊂ ∪ ∈ ∈ ⊂ ∈ ∈∈ ∪ ∈ ∈ ∈

−

= =

−

= =

=

= + + + + + +

= − + + + + + + + + +

+ − + + − + + + + − +

+ + + + + + + +

∑ ∑ ∑ ∑ ∑ ∑

∑ ∑

∑ ∑

( )( )2 2

02 1 .

a c

ia c i

−

=

+ − + +

∑ ∑


( )( )( ) ( ) ( ) ( ) ( )

( ) ( )

( )( )

( ) ( )( )( )

32 22 2

3 32 2 2 2 2

2 2

0 0

2 2

0 0

, ,

100 198 1630 24 4 1 8 1 18 1 8 13 3 3

20 a-1 134c 28c+ +20ac +8a 8 4 c+6c -8ac -4 a-1 c +3 3 3

1 2 3 1 2 2 6 6

2 2 2 3 2 2

b a a

i j

b a c

i j

W G a b c

aa aa a a a a a a a

c bc b

i j

a c a c i j

−

= =

−

= =

=

− + + + + + − + − + − − − +

= +

+ + +

+ + + + + + + + +

+ − + + − + + + +

∑ ∑

∑ ∑ ( ) ( ) ( )

( ) ( )( )

32

2 2 2

0 0

26 161 12

3 3

1 2 3 4 5 2 3 4 3 2 1 .a c

i i

b a b ab a

i a c i−

= =

− − + + − +

+ + + + + + + + + − + +

∑ ∑

After some algebraic manipulations, we get



( )( )3 3

2 2 2 2

32 2 2 2 2

97 20 146 16 149, , 35 4 20 8 28 83 3 3 3 3

2812 4 20 8 4 4 8 .3

a a b b cW G a b c a ab a b b ab


= + + + + + − + + + + +

+ + + + + − − +

□

THE MAIN RESULT

From Lemmas 1–6 the main result follows. Theorem 1: Let , , .a b c N∈ Then

( )( )

32 2

32 2

32 2 2 2 2

32 2

32 2

2

98 16 10030 24 4 83 3 3

20b 134a18 8 + + + ,3 3

28a+20ac +8c 8 4 a+6a -8ca -4ba +3

97 20 14635 4 20 83 3 3

16 14928 8 123 3

4 20

, ,

c c bc cb c b

b cb a b c

a ba b

c c bc cb c b

b ab cb ac

c a b

W G a b c

+ + + + + + +

+ − ≤ < + +

+ + + + + − +

+ + + + + +

+ +

=

32 2 2 2

2 3

32 2 2

3 32 2 2

2 2

,

288 4 4 83

14631 82 20 16 403 ,

1616 28 163

110 16 52 11026 20 16 16 32 643 3 3 3 3

16 16 16 64 32

a c b

aa b a a ca ba

ba a a aba c b

ba b b ab

c c b b ac cb b cb

ac c a ba abc b a

< ≤ + + − − +

+ + + + + − = ≤ − + + +

+ + + + + − + − + +

+ + − + +3

2 2

3 32 2 2

32 2 2 2

3 32 2 2 2

,1620 16

3110 16 52 11026 20 16 16 32 64

3 3 3 3 3 ,1616 16 16 64 32 20 16

397 20 146 1635 4 20 8 28 8

3 3 3 3

b a caa ca

a a b b ca ab b abb c a


a a b ba ab a b b ab

< ≤

+ +

+ + + + + − + − + + < <

+ + − + + + +

+ + + + + − + + + +

+3

2 2 2 2 2

3 32 2 2 2

32 2 2 2 2

,149 2812 4 20 8 4 4 8

3 398 16 100 20b30 24 4 8 18 8 + +

3 3 3 3 ,134c 28c+ +20ac +8a 8 4 c+6c -8ac -4bc +

3 3

c a bc cac a c bc b c c ac bc

a a ba ab a b b abc b a

c bc b

< ≤

+ + + + + − − +

+ + + + + + + − ≤ <

+ +

□



APPLICATIONS

Computer program based on the theorem Here we give a program written in Mathematica that uses the theorem from above to calculate values of

G(a,b,c).

[ ]

[ ]

3 32 2 2

32 2 2 2

3 32 2 2 2

2 2 2

110 16 52 1101 _, _, _ 26 20 16 16 32 643 3 3 3 3

1616 16 16 64 32 20 163

98 16 100 20b 134c2 _, _, _ 30 24 4 8 18 8 + + +3 3 3 3 3

+20ac +8a 8 4 c+6c -

a a b b cg a b c a ab b ab


a a bg a b c a ab a b b ab

c bc b

= + + + + + − + − + +

+ + − + + + +

= + + + + + + + −

+ +

[ ]

[ ]

[ ] [

32 2

32 3 2 2 2

3 32 2 2 2

32 2 2 2 2

28c8ac -4bc +3

146 163 _, _, _ 31 82 20 16 40 16 28 163 3

97 20 146 16 1493 _, _, _ 35 4 20 8 28 83 3 3 3 3

2812 4 20 8 4 4 83

_, _, _

b bg a a b c a a a ab a b b ab

a a b b cg b a b c a ab a b b ab


g a b c Which

= + + + + + − + + +

= + + + + + − + + + + +

+ + + + + − − +

=

[ ][ ][ ]

[ ][ ]

[ ][ ] ]

{ }

&& , 2 , , ,

&& , 3 , , ,

&& , 3 , ,

&& , 1 , , ,

&& , 1 , , ,

&& , 3 , , ,

&& , 2 , ,

, 5,5,5

a b b c g c b a

a c c b g b c b a

a c c b g a a b

b a a c g c b a

b c c a g a b c

c a a b g b a b c

c b b a g a b c

ar Array g

<= <

< <=

== <=

< <=

< <

< <=

<= <

=

Calculations

Using this program we calculated values for ( ), , , 1 , , 5G a b c a b c≤ ≤ . These values are given below for five values of a:



1\ 1 2 3 4 51 271 440 782 1300 20262 513 652 944 1498 22763 907 1082 1301 1756 25704 1485 1712 1967 2294 29485 2279 2574 2881 3244 3703

ab c=

2\ 1 2 3 4 51 440 654 1068 1682 25282 652 815 1140 1746 25923 1082 1281 1524 2020 29024 1712 1963 2242 2597 33045 2574 2893 3224 3615 4110

ab c=

3

\ 1 2 3 4 51 782 1068 1594 2352 33742 944 1140 1510 2188 31303 1301 1524 1791 2320 32544 1967 2242 2545 2924 36725 2881 3224 3579 3994 4517

ab c=

4\ 1 2 3 4 51 1300 1682 2352 3286 45162 1498 1746 2188 2978 40643 1756 2020 2320 2894 39004 2294 2597 2924 3327 41085 3244 3615 3994 4433 4980

ab c=

5\ 1 2 3 4 51 2026 2528 3374 4516 59862 2276 2592 3130 4064 53263 2570 2902 3254 3900 50184 2948 3304 3672 4108 49345 3703 4110 4517 4980 5551

ab c=



Use of the ring-matrix

Let us describe the notion of the ring-matrix of a benzenoid graph by the following example. For the following multiply-connected benzenoid graph, that is, a benzenoid graph with a hole [26]:

and the corresponding ring-matrix is given by:

0 0 1 1 1 1 1 0 0 01 1 1 1 0 0 0 1 0 00 0 1 1 1 0 0 1 0 01 1 1 1 1 1 1 1 1 00 1 1 1 1 1 0 0 0 00 1 1 1 1 1 1 1 1 1

The connection between the ring-matrix and a benzenoid graph is illustrated below:

111111111

1 1 1 1 1

111111111

1 1 1 1

1 1 1 1

1 1 1 1 1

1

0

0 0 0 0 0

0 0

0 0

0 0

0

0

0

0

0 0

0 0

0 0 0

1 1 1

11 11 11

11 11 11 11 11

11

11

1111111111111111 11

11 11 11 11 11

1111 11 11 11 11 11 11 11

Here we give a program, written in a Pascal-based pseudocode, that calculates Wiener index of a ben-

zenoid system from its ring-matrix. We have used this program to verify our formulas. The results given by this program coincide with the results given in subsection 4.2. Here is the program:



1. Read dimensions of the ring-matrix and store ring-matrix in array a (locations a(1,1)-a(na,ma)). 2. Add columns a(x,0) and a(x,ma+1), 0<=x<=na+1 consisting of zeros and add rows a(0,y) and

a(na+1,y) consisting of zeros. 3. Set all entries of array g to zero. 4. For each 1<=i<=na+1 and 1<=j<=2*ma+2 do if ( ((i%2) == 1) AND ((j%2) == 1) ) {symbols “a%b” here and forward mean “a mod b” } BEGIN if ( (a[i - 1][j/2] == 0) AND (a[i][j/2] == 0) AND (a[i][j/2 + 1] == 0) ) {symbols “a/b” here and

forward mean integer division} g[(i - 1) * ( 2 * ma + 2) + j][(i - 1) * (2 * ma + 2) + j] = 2;

else BEGIN if ( a[i-1][j/2] ) BEGIN if ( j > 1 ) g[(i - 1) * (2 * ma + 2) + j][(i - 1) * (2 * ma + 2) + j - 1] = 1; if ( j < 2 * ma + 2 ) g[(i - 1) * (2 * ma + 2) + j][(i - 1) * (2 * ma + 2) + j + 1] = 1; END if ( a[i][j/2] ) BEGIN if ( j > 1 ) g[(i - 1) * (2 * ma + 2) + j][(i - 1) * (2 * ma + 2) + j - 1] = 1; if ( i < na + 1 ) g[(i - 1) * (2 * ma + 2) + j][i * (2 * ma + 2) + j] = 1; END if ( a[i][j/2 + 1] ) BEGIN if ( j < 2 * ma + 2 ) g[(i - 1) * (2 * ma + 2) + j][(i - 1) * (2 * ma + 2) + j + 1] = 1; if ( i < na + 1 ) g[(i - 1) * (2 * ma + 2) + j][i * (2 * ma + 2) + j] = 1; END END END if ( ((i%2) == 1) AND ((j%2) == 0) ) BEGIN if ( (a[i][j/2] == 0) AND (a[i-1][j/2 - 1] == 0) AND (a[i-1][j/2] == 0) ) g[(i - 1) * ( 2 * ma + 2) + j][(i - 1) * (2 * ma + 2) + j] = 2; else BEGIN if ( a[i][j/2] ) BEGIN



if ( j < 2 * ma + 2 ) g[(i - 1) * ( 2 * ma + 2) + j][(i - 1) * (2 * ma + 2) + j + 1] = 1; if ( j > 1 ) g[(i - 1) * ( 2 * ma + 2) + j][(i - 1) * (2 * ma + 2) + j - 1] = 1; END if (a[i-1][j/2 - 1] ) BEGIN if ( j > 1 ) g[(i - 1) * ( 2 * ma + 2) + j][(i - 1) * (2 * ma + 2) + j - 1] = 1; if ( i > 1 ) g[(i - 1) * ( 2 * ma + 2) + j][(i - 2) * (2 * ma + 2) + j] = 1; END if (a[i-1][j/2] ) BEGIN if ( j < 2 * ma + 2 ) g[(i - 1) * ( 2 * ma + 2) + j][(i - 1) * (2 * ma + 2) + j + 1] = 1; if ( i > 1 ) g[(i - 1) * ( 2 * ma + 2) + j][(i - 2) * (2 * ma + 2) + j] = 1; END END END if ( ((i%2) == 0) AND ((j%2) == 1) ) BEGIN if ( (a[i][j/2] == 0) AND (a[i-1][j/2] == 0) AND (a[i-1][j/2 + 1] == 0) ) g[(i - 1) * ( 2 * ma + 2) + j][(i - 1) * (2 * ma + 2) + j] = 2; else BEGIN if ( a[i][j/2] ) BEGIN if ( j < 2 * ma + 2 ) g[(i - 1) * ( 2 * ma + 2) + j][(i - 1) * (2 * ma + 2) + j + 1] = 1; if ( j > 1 ) g[(i - 1) * ( 2 * ma + 2) + j][(i - 1) * (2 * ma + 2) + j - 1] = 1; END if (a[i-1][j/2] ) BEGIN if ( j > 1 ) g[(i - 1) * ( 2 * ma + 2) + j][(i - 1) * (2 * ma + 2) + j - 1] = 1; if ( i > 1 ) g[(i - 1) * ( 2 * ma + 2) + j][(i - 2) * (2 * ma + 2) + j] = 1; END

if (a[i-1][j/2 + 1] ) BEGIN if ( j < 2 * ma + 2 )



g[(i - 1) * ( 2 * ma + 2) + j][(i - 1) * (2 * ma + 2) + j + 1] = 1; if ( i > 1 ) g[(i - 1) * ( 2 * ma + 2) + j][(i - 2) * (2 * ma + 2) + j] = 1; END END END if ( ((i%2) == 0) AND ((j%2) == 0) ) BEGIN if ( (a[i - 1][j/2] == 0) AND (a[i][j/2 - 1] == 0) AND (a[i][j/2] == 0) ) g[(i - 1) * ( 2 * ma + 2) + j][(i - 1) * (2 * ma + 2) + j] = 2; else BEGIN if ( a[i-1][j/2] ) BEGIN if ( j > 1 ) g[(i - 1) * (2 * ma + 2) + j][(i - 1) * (2 * ma + 2) + j - 1] = 1; if ( j < 2 * ma + 2 ) g[(i - 1) * (2 * ma + 2) + j][(i - 1) * (2 * ma + 2) + j + 1] = 1; END if ( a[i][j/2 - 1] ) BEGIN if ( j > 1 ) g[(i - 1) * (2 * ma + 2) + j][(i - 1) * (2 * ma + 2) + j - 1] = 1; if ( i < na + 1 ) g[(i - 1) * (2 * ma + 2) + j][i * (2 * ma + 2) + j] = 1; END if ( a[i][j/2] ) BEGIN if ( j < 2 * ma + 2 ) g[(i - 1) * (2 * ma + 2) + j][(i - 1) * (2 * ma + 2) + j + 1] = 1; if ( i < na + 1 ) g[(i - 1) * (2 * ma + 2) + j][i * (2 * ma + 2) + j] = 1; END END END 5. Eliminate all colums and rows from g that contain entry 2. 6. Dijkstra's algorithm [27] is used to calculate distances between vertices. 7. Distances between vertices are summed up.

REFERENCES

[1] S. Nikolić, N. Trinajstić and Z. Mihalić, The Wiener in-dex: development and application, Croat. Chem. Acta, 68, 105–129 (1995).

[2] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc., 69, 17–20 (1947).

[3] D. H. Rouvray and R. B. King, eds., Topology in Chemistry – Discrete Mathematics of Molecules, Horwood, Chichester, 2002.

[4] Z. Mihalić, D. Veljan, D. Amić, S. Nikolić, D. Plavšić, N. Trinajstić, The distance matrix in chemistry, J. Math. Chem., 11, 223–258 (1992).

[5] H. Hosoya, A newly proposed quantity characterizing the topological nature of structural isomers of saturated hy-drocarbons, Bull. Chem. Soc. Japan, 44, 2332–2339 (1971).

[6] N. Trinajstić, ed., Mathematics and computational con-cepts in chemistry, Horwood/Wiley, New York, 1986.

[7] A. T. Balaban, ed., From chemical topology to three-dimensional geometry, Plenum Press, New York, 1997.

[8] J. Devillers and A.T. Balaban, eds., Topological indices and related descriptors in QSAR and QSPR, Gordon and Breach Sci. Publ., Amsterdam, 1999.

[9] R. Todeschini and V. Consonni, Handbook of molecular descriptors, Wiley-VCH, Weinheim, 2000, pp. 497–502.

[10] M. V. Diudea, ed., QSPR/QSAR studies by molecular descriptors, Nova Sci. Publ. Huntington, NY., 2001.

[11] I. Gutman, D. Rouvray, A new theorem for the Wiener molecular branching index of trees with perfect match-ings, Comput. Chem., 14, 29–32 (1990).

[12] A. A. Dobrynin, Graphs of unbranched hexagonal sys-tems with equal values of the Wiener index and different numbers of rings, J. Math. Chem., 9, 239–252 (1992).

[13] A. A. Dobrynin, A new formula for the calculation of the Wiener index of hexagonal chains, MATCH – Comm. Math. Comp. Chem., 35, 75–90 (1997).

[14] A. A Dobrynin, Formula for calculating the Wiener index of catacondensed benzenoid graphs, J. Chem. Inf. Com-put. Sci., 38, 811–814 (1998).

[15] A. A. Dobrynin, Explicit relation between the Wiener index and the Schultz index of catacondensed benzenoid graphs, Croat. Chem. Acta, 72, 869–874 (1999).

[16] A. A. Dobrynin, A simple formula for the calculation of the Wiener index of hexagonal chains, Comput. Chem., 72, 869–874 (1999).

[17] A. A. Dobrynin, I. Gutman, The average Wiener index of hexagonal chains, Comput. Chem., 23, 571–576 (1999).

[18] A. A. Dobrynin, I. Gutman, S. Klavžar and P. Žigert, Wiener index of hexagonal systems, Acta Appl. Math., 72, 247–294 ( 2002).

[19] I. Gutman and S. J. Cyvin, Introduction to the Theory of Benzenoid Hydrocarbons, Springer-Verlag, Berlin, 1989.

[20] N. Trinajstić, On classification of polyhex hydrocarbons, J. Math. Chem., 5, 171–175 (1990).

[21] N. Trinajstić, Chemical Graph Theory, CRC Press, Boca Raton, FL., 1992, Ch.10.

[22] J. R. Dias, Handbook of polycyclic hydrocarbons. Part A. Benzenoid hydrocarbons, Elsevier, Amsterdam, 1987.

[23] R. A. Hites and W. J. Simonsick, Jr., Calculated molecu-lar properties of polycyclic aromatic hydrocarbons, El-sevier, Amsterdam, 1987.

[24] R. G. Harvey, Polycyclic aromatic hydrocarbons, Wiley-VCH, New York, 1997.

[25] S. Klavžar, I. Gutman, A. Rajapakse, Wiener numbers of pericondensed benzenoid hydrocarbons, Croat. Chem. Acta, 70, 979–999 (1997).

[26] N. Trinajstić, On the classification of polyhexes, J. Math. Chem., 9, 373–380 (1992).

[27] A. Gibbons, Algorithmic graph theory, Cambridge Uni-versity Press, Cambridge, 1985.

R e z i m e

VINEROVI INDEKSI NA BENZENOIDNI GRAFOVI

Damir Vukičević1, Nenad Trinajstić2

1Zavod za matematiku, Sveučilište u Splitu, Teslina 12, HR-21000 Split, Hrvatska

2Institut "Ru|er Bošković", p.p. 180, HR-10001 Zagreb, Hrvatska

[email protected]

Klu~ni zborovi: benzenoidni grafovi; perikondenzirani benzenoidi; prsteni-matrici; Vinerov indeks

Izvedeni se formuli za Vinerovi indeksi na klasa od perikondenzirani benzenoidni grafovi koi se sostojat od tri reda na {estoagolnici so razli~na dol`ina. Za taa cel e podgotvena i soodvetna kompju-terska programa. Dobienite rezultati se provereni i sporedeni so broevite presmetani so psevdokod ori-entiran vo programskiot paket Paskal, koj e baziran na prsteni-matrici na benzenoidni grafovi. Ovaa

programa mo`e da presmeta Vinerov indeks na koj i da e benzenoiden sistem, ne samo za perikondenzirani benzenoidi koi se razgledani vo ovoj trud.

Documents

WIENER INDICES OF BENZENOID GRAPHS* · chemistry [1]. It was introduced in 1947 by Harold Wiener, then a chemistry student at the Brooklyn College, as the path number for characterization