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Inequalities Involving the Inequalities Involving the Coefficients of Independence Coefficients of Independence
PolynomialsPolynomials
Combinatorial and Probabilistic Inequalities Isaac Newton Institute for Mathematical
Sciences Cambridge, UK - June 23-27, 2008
Vadim E. LevitVadim E. Levit1,21,2
11Ariel University Center of Samaria, Ariel University Center of Samaria, Israel Israel
Eugen MandrescuEugen Mandrescu22
22Holon Institute of Technology, Holon Institute of Technology,
IsraelIsrael
II(G;x)(G;x) = = the independence the independence
polynomialpolynomial of graph of graph GGResults and conjectures onResults and conjectures on
II(G;x)(G;x) for some graph for some graph classes…classes…
…… some some moremore open open problemsproblems
O u t l i n eO u t l i n e
IntroductionIntroduction
Quasi-Regularizable GraphsQuasi-Regularizable Graphs
König-EgervKönig-Egervááry Graphsry Graphs
The Main InequalityThe Main Inequality
Well-Covered GraphsWell-Covered Graphs
Perfect GraphsPerfect Graphs
Corona GraphsCorona Graphs
Palindromic GraphsPalindromic Graphs
A set ofA set of pairwise non-adjacent pairwise non-adjacent verticesvertices is called ais called a stable setstable set or anor an
independent setindependent set..
(GG) = stability number is the
maximum size of a stable set of GG.
Some Some definitionsdefinitions
Ga
eb
c
d Stable sets in G : , , {a}, {a,b}, {a, b, c}, {a}, {a,b}, {a, b, c},
{a, b, c, d} , … {a, b, c, d} , …
(GG) = |{a,b,c,d,e,fa,b,c,d,e,f}| = 5
ExampleExample
IfIf sskk denotes the number of denotes the number of stable setsstable sets of size of size kk in a graphin a graph
GG withwith (G) = (G) = ,, thenthen
II((GG) = ) = II((GG;;xx) = ) = ss00 ++ ss11xx ++……++ ssxx
is called theis called the
independence polynomialindependence polynomial ofof GG..
Gutman & Harary - ‘83Harary I. Gutman, F. Harary , I. Gutman, F. Harary ,
Generalizations of matching polynomialGeneralizations of matching polynomial Utilitas Mathematica 24 (1983)Utilitas Mathematica 24 (1983)
AllAll the the stable setsstable sets ofof GG : : …… …… {a}, {b}, {c}, {d}, {e}, {f}{a}, {b}, {c}, {d}, {e}, {f} …….. .. {a, b}, {a, d}, {a, e}, {a, f}, {a, b}, {a, d}, {a, e}, {a, f}, {b, c}, {b, e}, {b, f}, {d, f}, {b, c}, {b, e}, {b, f}, {d, f}, ………… { a, b, e{ a, b, e }, { a, b, f}, { a, b, f }, { a, d, f }}, { a, d, f }
1 6 8
3
G c
b
d
a
f
e I(G) = 1 + 6x + 8x2 + 3x3
Example
ALSO non-isomorphic trees can have the same independence
polynomial !
T1
T2
K. Dohmen, A. Ponitz, P. Tittmann, Discrete Mathematics and Theoretical Computer Science 6 (2003)
I(T1) = I(T2) = 1+10x+36x2+58x3+42x4+12x5++x6
E = E = { a, b, c, d, e, f } { a, b, c, d, e, f }
The The line graphline graph of of G = (V,E)G = (V,E) is is LG = LG = (E,(E,UU)) where where ababUU whenever the whenever the
edges edges a, ba, bEE share ashare a common vertex in G.
…… for historical reasonsfor historical reasons
G a
e
b
f
c d
{a, b, d} = matchingmatching in G {a, b, d} = stable set in LG
LGa
b
e f
cd
Example
If If GG has has nn vertices, vertices, mm edges, and edges, and mmkk
matchings ofmatchings of sizesize kk,, thenthen thethematching polynomialmatching polynomial ofof GG is is
whilewhile
is the is the positive matching polynomialpositive matching polynomial ofof GG..
knk
m
k
k xmxGM 2
0
)1();(
…… recall for historical reasonsrecall for historical reasons
I. Gutman, F. Harary,I. Gutman, F. Harary, Generalizations of matching polynomialGeneralizations of matching polynomial Utilitas Mathematica 24 (1983) Utilitas Mathematica 24 (1983)
If If GG has has nn vertices, vertices, mm edges, and edges, and mmkk
matchings ofmatchings of sizesize kk,, thenthen
km
kk xmxGM
0
);(
Independence polynomial is a generalization of the matching
polynomial, i.e., M+(G;x) = I(LG; x), where LG is the line
graph of G.
M+(G;x) = I(LG;x) = 1+6x+7x2+1x3
G
c
b
d a
f e LGc
b
d
a f
e
G = (V,E) LG = (E,U)
Example
““Clique polynomialClique polynomial””: C(G;: C(G;xx) = ) = II((HH;-;-xx),), where where HH is the is the complement of complement of G,G, GoGoldwurm & Santini - 2000wurm & Santini - 2000
Twin:Twin: - - ““Independent set polynomialIndependent set polynomial”” Hoede & Li - 1994Hoede & Li - 1994
Some “relatives” of I( ; ) :Some “relatives” of I( ; ) :
““Dependence polynomialDependence polynomial”” : D(G; : D(G;xx) = ) = II((HH;-;-xx),), where where HH is the complement of is the complement of GG
Fisher & Solow - 1990Fisher & Solow - 1990
““Clique polynomialClique polynomial””: C(G;: C(G;xx) = ) = II((HH;;xx),), where where HH is the is the complement of complement of GG
Hajiabolhassan & Mehrabadi - 1998Hajiabolhassan & Mehrabadi - 1998
““Vertex cover polynomial of a graphVertex cover polynomial of a graph”, ”, where the coefficient where the coefficient aakk is the number of vertex covers V’ of G with |V’| = k is the number of vertex covers V’ of G with |V’| = k,,
Dong, Hendy & Little - 2002Dong, Hendy & Little - 2002
Chebyshev polynomials of the first and second kind:
Connections with other polynomials:Connections with other polynomials:
Hermite polynomials:
)2
(2; 2 xHxKLI n
n
n
I. Gutman, F. Harary, Utilitas Mathematica 24 (1983)I. Gutman, F. Harary, Utilitas Mathematica 24 (1983)
G. E. Andrews, R. Askey, R. Roy, Special functions (2000)G. E. Andrews, R. Askey, R. Roy, Special functions (2000)
)2(2; 2/11)1( xTxxCI nn
n
)2(14
2; 2/11)2(
2
2
xTx
xxPI n
n
n
P(G; x, y)P(G; x, y) is equal to the number of vertex colorings is equal to the number of vertex colorings : V : V {1; 2; …, x} {1; 2; …, x} of the graph of the graph G = (V,E)G = (V,E) such that such that
for all edges for all edges uv uv E E the relations the relations(u) (u) y y and and (v) (v) y y imply imply (u) (u) (v). (v).
The generalized chromatic polynomial : P(G;x,y)P(G;x,y)
K. Dohmen, A. Ponitz, P. Tittmann, K. Dohmen, A. Ponitz, P. Tittmann, A new two-variable A new two-variable generalization of the chromatic polynomialgeneralization of the chromatic polynomial, Discrete , Discrete
Mathematics and Theoretical Computer Science 6 (2003) 69-90.Mathematics and Theoretical Computer Science 6 (2003) 69-90.
P(G; x, y)P(G; x, y) is a polynomial in variables is a polynomial in variables x, yx, y, which , which simultaneously generalizes the simultaneously generalizes the chromatic polynomialchromatic polynomial, ,
the the matching polynomialmatching polynomial, and the , and the independenceindependencepolynomialpolynomial of of GG, e.g., , e.g., II(G; x) = P(G; x + 1, 1).(G; x) = P(G; x + 1, 1).
Connections with other polynomials:Connections with other polynomials:
RReemmararkk
How to compute the independence How to compute the independence polynomial ?polynomial ?
where G+H = (V(G)V(H);E) E = E(G)E(H){uv:uV(G),vV(H)}
2.I(G+H) = I(G) + I(H) – 1
If V(G)V(H) = , then
1. I(GH) = I(G) I(H)
GH = disjoint union of G and H
G+H = Zykov sum of G and H
I. Gutman, F. Harary, Utilitas Mathematica 24 (1983)I. Gutman, F. Harary, Utilitas Mathematica 24 (1983)
How to compute the independence How to compute the independence polynomial ?polynomial ?The The coronacorona of the graphs of the graphs GG andand HH is is
the graph the graph GG○○HH obtained from obtained from GG and and n = n = |V(G)||V(G)| copies of copies of HH, so that each vertex , so that each vertex of of GG is joined to all vertices of a copy is joined to all vertices of a copy
of of HH..
G
HH GG○○HH
GG
HH HH HH Example
I(GG○○H;H;xx) = (I(H;H;xx))n I(G; x x / I(H;xx))
I. Gutman, Publications de lI. Gutman, Publications de l’’Institute Mathematique 52 (1992)Institute Mathematique 52 (1992)
TheoreTheoremm
G
HH
GG○○HH I(G) = 1+3x+x2
I(H) = 1+2x
I(GG○○H;H;x) = (1+2x)3 I(G;x / (1+2x)) = = 1 + 9x + 25x2 + 22x3
Example
IfIf G = (V,E),G = (V,E), vvVV andand uvuv EE, then , then the following assertions are true:the following assertions are true:
PropositionProposition
wherewhere N(N(vv)) = {= { uu : : uuvv EE } } isis thethe neighborhoodneighborhood ofof v v V V andand N[v] = N(v) N[v] = N(v)
{v}.{v}.
(ii)(ii) II(G) = (G) = II(G (G –– uvuv) ) –– xx22 II(G (G –– N( N(uu))N(N(vv)),)),
(i)(i) II(G) = (G) = II(G (G –– vv) + ) + xx II(G (G –– N[ N[vv])])
I. Gutman, F. Harary, Utilitas Mathematica 24 (1983)I. Gutman, F. Harary, Utilitas Mathematica 24 (1983)
N[v] = {a,c,d}{v}
c
G a
d b
v
I(G) = I(G-v) + x I(G-N[v]) =
P4 a
c
d
b
= I(P4) + x I({b}) =
= 1 + 4 x + 3 x2 + x (1+x) =
I(P4) = 1 + 4 x + 3 x2
= 1 + 5 x + 4 x2
G-v = P4
G-N[v] = {b}
Example
Some properties of the Some properties of the coefficients of coefficients of
independence polynomial, independence polynomial, as …as …
- - unimodality unimodality - - log-concavitylog-concavity - - palindromicity …palindromicity …
- - definitions & examplesdefinitions & examples - - results & conjectures … results & conjectures …
(-1, 2, -3, 4) is NON-unimodal, but it is
log-concave: (-1)(-3) 22, 24 (-3)2
A sequence of reals aa00, a, a11,..., a,..., ann is: (i) unimodalunimodal ifif aa0 0 aa1 1 ... ... aamm ... ...
a an n
for somefor some mm{0,1,...,n}{0,1,...,n},,
(ii) log-concavelog-concave ifif aak-1k-1 a ak+1k+1 (a (akk))22
for every for every k k {1,...,n-1}. {1,...,n-1}.(1, 4, 5, 2) is both uni & log-con (1, 2, 5, 3) is unimodal, NON-log-concave: 15
> 22
However, every log-concavelog-concave sequence of positive numberspositive numbers is
unimodalunimodal..
Examples
A polynomialA polynomial
P P ((xx)) = = aa0 0 + + aa11xx + +……++ aannxxnn
is is unimodalunimodal (log-concave)(log-concave) if its if its sequencesequence of of coefficientscoefficients aa0 0 , a, a1 1 , ,
aa2 2 , ... , a, ... , ann
is is unimodalunimodal (log-concave, (log-concave, respectively).respectively).
P(x) = 1 + 4x + 50x2 + 2x3 is unimodal with mode k
= 2P(x) = (1 + x)n is unimodal with the
mode k = n/2 and is also log-concave
Example
Is there a (connected) graph G with (G) =
whose sequence
s0, s1, s2 , … , s
is NOT unimodal ?
Recall that sk denotes the number of stable sets of size k
in a graph.Question
H. Wilf
Answer
For 3, there is a (connected) graph G with (G) = whose sequence s0,
s1, s2 , … , s
is NOT unimodal ! Y. Alavi, P. Malde, A. Schwenk, P. ErdY. Alavi, P. Malde, A. Schwenk, P. Erdöös s
Congressus Numerantium 58 (1987)Congressus Numerantium 58 (1987)
II((HH) = ) = 11++6464 xx + +634 634 xx22 ++500500 xx3 ++625625 xx4 is is notnot unimodal unimodal
I(G) = 1 + 6x + 8x2 + 2x3 is unimodalunimodal
Examples
K5K22
K22K5
K5
K5
H
G
For any permutation of the set {1, 2, {1, 2, ……, , }},, there is a graph GG
such that (G) = (G) = and ss(1)(1)< < ss(2)(2)< < ss(3)(3)< < …… < < ss(())
where sskk is the number of stable stable setssets in GG of size kk..
Moreover, any deviation from unimodality is possible!
Theorem
Y. Alavi, P. Malde, A. Schwenk, P. ErdY. Alavi, P. Malde, A. Schwenk, P. Erdöös s Congressus Numerantium 58 (1987)Congressus Numerantium 58 (1987)
A graph is called claw-free if it has no clawclaw, ( i.e., K1,3 ) as an induced
subgraph. K1,3Theorem
II((GG)) is log-concave for is log-concave for every claw-free graph every claw-free graph G.G.
RemarkThere are non-claw-free graphs with log-concave independence
polynomial.
I(K1,3 ) = 1 + 4x + 3x2 + x3
Y. O. Hamidoune Y. O. Hamidoune Journal of Combinatorial Theory B 50 (1990)Journal of Combinatorial Theory B 50 (1990)
IfIf all the rootsall the roots of a of a polynomialpolynomial withwith positive positive
coefficientscoefficients areare realreal, , then the then the polynomial ispolynomial is log-concavelog-concave..Sir Sir II. Newton , . Newton , Arithmetica UniversalisArithmetica Universalis (1707) (1707)
Theorem
Moreover
,I(G) has only real roots, for every claw-free graph G.
Theorem
M. Chudnovsky, P. Seymour, J. Combin. Th. B 97 (2007)
IfIf TT is a tree, is a tree, thenthen II(T)(T) is unimodalis unimodal..
I(T) = 1+7x + 15x2 +14x3 +6x4 +x5
Still open …
Example
Conjecture 1Conjecture 1
Y. Alavi, P. Malde, A. Schwenk, P. ErdY. Alavi, P. Malde, A. Schwenk, P. Erdöös s Congressus Numerantium 58 (1987)Congressus Numerantium 58 (1987)
T
IfIf FF is a forest, is a forest, then then II(F)(F) is unimodalis unimodal..
I(F) = I(K1,3 ) I(P4) = 1+8x+22x2+25x3+13x4+3x5
F
Still open …
Example
Conjecture 2Conjecture 2
Y. Alavi, P. Malde, A. Schwenk, P. ErdY. Alavi, P. Malde, A. Schwenk, P. Erdöös s Congressus Numerantium 58 (1987)Congressus Numerantium 58 (1987)
There existThere exist unimodal independence polynomials whose product is not
unimodal.
I(GG) = I(G) I(G) = == 1++232x + + 13750x2 + + 34790x3
+ + 101185x4 + + 100842x5 + + 117649x6
I(G) = 1+116 x +147 x2+343 x3
Example G
K7 K95K7
K7
G = K95+3K7
(i)(i) log-concave log-concave unimodal =unimodal = unimodal;unimodal;
i.e.,i.e., log-concavelog-concave unimodalunimodal is notis not necessarily necessarily log-concavelog-concave
J. Keilson, H. Gerber J. Keilson, H. Gerber Journal of American Statistical Association 334 (1971)Journal of American Statistical Association 334 (1971)
Theorem
G = K40 + 3K7 , H = K110 + 3K7
P1 = I(G) = 1+61x +147x2+343x3 … log-concave
P2 = I(H) = 1+131x+147x2+343x3 … not log-con
P1 P2= 1+192x +8285x2+28910x3+ +87465x4+100842x5+117649x6
1008422 – 87465 117649 = – 121 060 821
HoHoweweveverr
(ii)(ii) log-concavelog-concave log-concavelog-concave = = log-log-concave.concave.
The unimodality of The unimodality of independence polynomials independence polynomials
ofof trees trees does does notnot directlydirectly implies implies
the unimodality of the unimodality of independence polynomials independence polynomials
ofof forestsforests! !
Consequence
Hence,Hence,
independence polynomials independence polynomials ofof forests are log-forests are log-concaveconcave as well ! as well !
IfIf TT is a tree, is a tree, then then II(T)(T) is log-concaveis log-concave..
Conjecture 1*Conjecture 1*
IntroductionIntroduction
Quasi-Regularizable GraphsQuasi-Regularizable Graphs
König-EgervKönig-Egervááry Graphsry Graphs
The Main InequalityThe Main Inequality
Well-Covered GraphsWell-Covered Graphs
Perfect GraphsPerfect Graphs
Corona GraphsCorona Graphs
Palindromic GraphsPalindromic Graphs
00 = 0, = 0, = n, = n, 11
G
-k = 2 (-k )
1 = = 2
2 = 3 < 3 = 6
0 = 1 < = 2
Let G be a graph of order n with (G) = and 0 k . Then-k = max{n|N[S]| : S is stable, |S| = k}.
Examples
H
Def
IfIf GG is a graph withis a graph with (G) = andand (G) = , thenthen
(ii) s 1 s-1 s-1.
Theorem
(i) (k+1) sk+1 -k sk , 0 k
V. E. Levit, E. Mandrescu, Graph Theory in Paris: Proceedings of a Conference in Memory of C. Berge (2006)
LetLet HH = = (A(A,,BB,,E)E) be a bipartite graph be a bipartite graph withwith
XXAA X X is ais a kk-stable set in-stable set in GG (|X|=k)(|X|=k)
YYBB Y Y is ais a (k+1)(k+1)--stable set instable set in GG,,
andand XYXYEE X X YY
anyany YYBB hashas k+1k+1 kk-subsets-subsets ||EE|| = (k+1)s = (k+1)sk+1k+1
ifif XXAA andand vvV(G) V(G) N[X] N[X] X X{v}{v}BB
hencehence, , degdegHH(X) (X) -k-k andand
(k+1) s(k+1) sk+1k+1= = ||EE|| -k-k s skk
PPrrooooff
IntroductionIntroduction
Quasi-Regularizable GraphsQuasi-Regularizable Graphs
König-EgervKönig-Egervááry Graphsry Graphs
The Main InequalityThe Main Inequality
Well-Covered GraphsWell-Covered Graphs
Perfect GraphsPerfect Graphs
Corona GraphsCorona Graphs
Palindromic GraphsPalindromic Graphs
GG is calledis called quasi-regularizablequasi-regularizable ifif |S| |S| |N(S)| |N(S)| for each stable setfor each stable set
SS..
Quasi-regQuasi-reg
Non-Non-quasi-quasi-
regreg
Definition
C. Berge, Annals of Discrete Mathematics 12 (1982)C. Berge, Annals of Discrete Mathematics 12 (1982)
ExamplExampleses
IfIf GG is a quasi-regularizable is a quasi-regularizable graph of ordergraph of order nn = 2 = 2(GG) = 22,
thenthen
(ii)(ii) (k+1)(k+1) ssk+1k+1 2 ( 2 (-k)-k) sskk
(i)(i) -k-k 2 ( 2 (-k)-k)
(iii)(iii) sspp s sp+1p+1 … … s s-1-1 ss
wherewhere p = p = (2(2-1)/3-1)/3..
Theorem
V. E. Levit, E. Mandrescu, Graph Theory in Paris: Proceedings of a Conference in Memory of C. Berge (2006)
ProofProof ((i)) IfIf SS is stable and is stable and ||SS| = k| = k 2 |S| 2 |S| |S |SN(S)|N(S)| 2(2(-k) = 2(-k) = 2(-|S|) -|S|) n-|N[S]| n-|N[S]| -k -k 2 ( 2 (-k)-k)
(ii)(ii) (k+1) sk+1k+1 22 ((-k)-k) s sk k
becausebecause (k+1) sk+1k+1 -k-k skk
((iii)) ssp p … … s s-1 -1 s s forfor pp = =
(2(2-1)/3-1)/3 sincesince byby (ii),(ii), it follows it follows
thatthat ssk+1k+1 ssk k ,, whenever whenever
kk +1 +1 2( 2(-k) -k) p p = = (2(2-1)/3-1)/3
We found out that (sk) is decreasing in this upper part:
ifif GG is quasi-regularizable of orderis quasi-regularizable of order 22(G), , thenthen
1 2 3 p k
sk
decreasing
Unimodal ? Log-concave ?
Unconstrained ?
sp … s-1 s , p = p = (2(21)/3)1)/3)
ExampExamplele
G is quasi-regularizable
I(G) = 1 + 8 x + 19 x2 + 15 x3 + 4 x4
p = (2-1)/3 = (8-1)/3 = 3
(G) = 4 n = 8
G is quasi-regularizable & I(G) is log-concave
s3 = 15 s4 = 4
G
ExampExamplele
G is not a quasi-
reg graph
I(G) = 1 + 9 x + 26 x2 + 30 x3 + 17 x4 + 4 x5
p = (2-1)/3 = (10-1)/3 = 3
(G) = 5 n = 9
G is not quasi-regularizable & I(G) is log-concave
s3 = 30 s4 = 17 s5 =
4
G
ExampExamplele G is a
quasi-reg
graph
I(G) = 1 + 16 x + 15 x2 + 20 x3
+15 x4 + 6 x5 + 1 x6
p = (2-1)/3 = (12-1)/3 = 4
(G) = 6 n = 16
G is quasi-reguralizable & I(G) is not unimodal!
G = K10 + 6K1 s4 = 15 s5 = 6 s6 = 1
G K10
K1
K1K1
K1
K1K1
IntroductionIntroduction
Quasi-Regularizable GraphsQuasi-Regularizable Graphs
König-EgervKönig-Egervááry Graphsry Graphs
The Main InequalityThe Main Inequality
Well-Covered GraphsWell-Covered Graphs
Perfect GraphsPerfect Graphs
Corona GraphsCorona Graphs
Palindromic GraphsPalindromic Graphs
A graph G is called well-covered if all its
maximal stable sets are of the same size (namely, (G)).
If, in addition, G has no isolated vertices and its order equals
2(G), then G is called very well-covered.
M. L. Plummer, J. of Combin. Theory 8 (1970)
O. Favaron, Discrete Mathematics 42 (1982)
Definitions
C4 & H2 are very well-
covered
Examples H1H1 is well-covered
H4
H3 & H4 are
not well-covered
H2C4
H3
G is a well-covered graph, I(G) = 1+9x+ 25x2 +22x3 is
unimodal.
If GG is a well-covered graph, then II(G)(G) is unimodal.
ExEx-Conjecture -Conjecture 33
G
J. I. Brown, K. Dilcher, R. J. Nowakowski J. I. Brown, K. Dilcher, R. J. Nowakowski J. of Algebraic Combinatorics 11 (2004) J. of Algebraic Combinatorics 11 (2004)
Example
i.e., i.e., Conjecture 3Conjecture 3 is is truetrue for every well-covered for every well-covered
graphgraph GG havinghaving (G) (G) 3 3. .
They also provided
counterexamples for 4 (G) 7.
T. Michael, W. Traves, Graphs and Combinatorics 20 (2003)T. Michael, W. Traves, Graphs and Combinatorics 20 (2003)
Theorem I(I(GG) is unimodal) is unimodal for for
everyeverywell-covered graphwell-covered graph GG
havinghaving (G) (G) 3 3. .
K4, 4,…, 4
1701
K10K10
K10K10
GG = 4K10 + K4, 4, …, 4
1701 times
1701-partite: each part has 4 vertices
GG
Michael & Traves’ counter Michael & Traves’ counter exampleexample
G = 4KG = 4K10 10 ++ KK4, 4, 4, 4, ……, 4, 4
n times 4
GG is well-covered, (GG) = 4 I(G) = 1+(40+4n) x+(600+6n) x2 + (4000+4n) x3 +
(10000+n) x4
I(G) is NOT unimodal iff 1701 n 1999
and it is NOT log-concave iff 24 n 2452
I(G) is NOT unimodal iff 4000+4n min{40+4n,1000+n}
KK4, 4,…, 4
1701
KK10KK10
KK10KK10
K1000
K1000
K1000
K1000
GG1701 times
GG = (4K(4K10 + KK4,4,…,4)) (4K1000)
(GG) = 8
G is well-covered and (G) = 8, while
I(G) = 1 + 14,844 x + 78,762,806 x2
+ 196,342,458,804 x3
+ 235,267,430,443,701 x4
+ 109,850,051,389,608,000 x5
+ 173,242,008,824,000,000 x6
+ 173,238,432,000,000,000 x7
+ 187,216,000,000,000,000 x8
1701 times
G = (4K10+K4,4,…,4) qK1000q
GG isis well-well-coveredcovered
K4, 4,…, 4
1701
K1000q
K10
K1000q
K10
K10
K10
q times
((GG)) = q + = q + 44
0 0 ≤≤ q q
V. E. Levit, E. Mandrescu, European J. of Combin. 27 (2006)
qq = = 00
K4, 4,…, 4
1701
K1000q
K10
K1000q
K10
K10
K10
q times
4 4 ≤≤ q q
11 ≤ ≤ qq ≤ ≤ 33
Michael & Traves CounterExample
New Michael & Traves CounterExamples
CounterExamples forCounterExamples for 8 8 ≤≤
Gq is well-covered, not
connected, (Gq) = q + 4
I(Gq;x) = (1+ 6844 x + 10806 x2 +
10804 x3 + 11701 x4) (1 + 1000q x)q
is not unimodal. Proof: sq+2 > sq+3 < sq+4
G is very well-covered I(G) = 1 +
6x + 9x2 + 4x3
Example
Conjecture 3*
G
IfIf GG is a very well-covered
graph, then then II(G)(G) is unimodal.
V. E. Levit, E. Mandrescu, Graph Theory in Paris: Proceedings of a Conference in Memory of C. Berge (2006)
If G is a well-covered graph with (G) = , then
(i) 0 k
(-k) sk (k+1) sk+1
(ii) s0 s1… sk-1 sk , k = (+1)/2 .
Theorem
T. Michael, W. Traves, Graphs and Combinatorics 20 (2003)T. Michael, W. Traves, Graphs and Combinatorics 20 (2003)
V. E. Levit, E. Mandrescu, V. E. Levit, E. Mandrescu, Discrete Applied Mathematics 156 (2008) Discrete Applied Mathematics 156 (2008)
eacheach (k+1)--stable set includesstable set includes k+1
stable sets of sizestable sets of size k (k+1) sk+1
EveryEvery kk-stable set-stable set AAkk is included in is included in some stable setsome stable set BB of sizeof size ..
thus, (-k) sk (k+1) sk+1
hence, eachhence, each BB hashas -k-k stable subsetsstable subsets ofof
sizesize k+1k+1 that includethat include AAkk (-k) sk
sk-1 sk , for k (+1)/2
ProofProof
IfIf GG is a is a veryvery well-coveredwell-covered graph with graph with (G) = (G) = , then, then
(v) II(G)(G) is unimodal, whenever 9 9..
(iii) sspp s sp+1p+1 …… s s-1-1 s s,,p = p = (2(2-1)/3-1)/3
(ii) ss0 ss1 … ss /2
(i) ((-k) s-k) skk (k+1) s (k+1) sk+1 k+1 2 (2 (-k) s-k) skk
Theorem
V. E. Levit, E. Mandrescu, Graph Theory in Paris: Proceedings of a Conference in Memory of C. Berge (2006)
(iv) ss ss-2 (ss-1)2
(iv) CombiningCombining (ii) andand (iii),, it follows it follows that that II(G)(G) is unimodal, whenever is unimodal, whenever 99..
(i) It follows from previous results on It follows from previous results on quasi-reg graphs, as any well-covered quasi-reg graphs, as any well-covered graph is quasi-regularizable (Berge)graph is quasi-regularizable (Berge)
(i) (ii) s0 s1 … s/2
(i) (iii) ssp p s sp+1 p+1 …… s s-1 -1 s s
wherewhere p = p = (2(2-1)/3-1)/3
ProofProof
For anyany permutation of {k, k+1,…, }, k = /2, there is a well-coveredwell-covered graph G with (G) = , whose sequence
(s0 , s1 , s2 ,…, s) satisfies:
s(k)< s(k+1)< …< s().
Conjecture 4 : Conjecture 4 : “Roller-“Roller-Coaster”Coaster”
T. Michael, W. Traves, Graphs & Combinatorics 20 (2003)T. Michael, W. Traves, Graphs & Combinatorics 20 (2003)
The “The “Roller-CoasterRoller-Coaster” Conjecture” Conjecture isis validvalid forfor
What about (G) > (G) > 1111 ?
Still open …
T. Michael, W. Traves, Graphs and Combinatorics 20 (2003)T. Michael, W. Traves, Graphs and Combinatorics 20 (2003)
(i)(i) every well-covered graph GG with (G) (G) 7 7;;
(ii)(ii) every well-covered graph GG with (G) (G) 11. 11.
P. Matchett, Electronic Journal of Combinatorics (2004)P. Matchett, Electronic Journal of Combinatorics (2004)
For a well-covered graph, the sequence (sk) is unconstrained
with respect to order in its upper part!
1 2 3 2
k
sk
increasing
unconstrained
““Roller-CoasterRoller-Coaster” ” conjecture:conjecture:
P. Matchett (2004)P. Matchett (2004)
““Roller-CoasterRoller-Coaster”” conjecture*:conjecture*:For a VERY well-covered graph,
the sequence (sk) is unconstrained with respect to
order in this upper part!
1 2 3 2-1 3
k
sk
increasing
unconstrained
2
decreasing
V. E. Levit, E. Mandrescu (2006)
IntroductionIntroduction
Quasi-Regularizable GraphsQuasi-Regularizable Graphs
König-EgervKönig-Egervááry Graphsry Graphs
The Main InequalityThe Main Inequality
Well-Covered GraphsWell-Covered Graphs
Perfect GraphsPerfect Graphs
Corona GraphsCorona Graphs
Palindromic GraphsPalindromic Graphs
GG is calledis called perfectperfect ifif (H) = (H) = (H)(H) for any induced for any induced subgraphsubgraph HH ofof GG, , wherewhere (H), (H), (H)(H) are the are the chromatic andand thethe clique numbers of of HH..
C. Berge, 1961C. Berge, 1961
E.g.,E.g., any any chordal graph is chordal graph is
perfect.perfect.
IfIf GG is a perfect graphis a perfect graphwith with (G) = (G) = andand (G) = (G) = , then, then
ssp p s sp+1 p+1 …… s s-1 -1 s s
wherewhere p = p = (( 1) / ( 1) / ( 1) 1)..
= 3,= 3, = 3, p = 2= 3, p = 2
G
II(G)(G) = 1+6x+8x2+3x3
Theorem
Example
We found out that the sequence (sk) is decreasing in its upper part:
ifif GG is ais a perfect graphperfect graph withwith (G) = , (G) = , then then ssp p s sp+1 p+1 …… s s-1 -1 s s for p = =
((-1)/(-1)/(+1)+1)..
1 2 3 -1+1
k
sk
decreasing
Unimodal ? Log-concave ?
Unconstrained ?
IfIf SS is stable andis stable and ||SS| = | = kk, , thenthen H = G-N[S]H = G-N[S] hashas ((HH) ) ((GG)-)-kk..
By Lovasz’s theoremBy Lovasz’s theorem ||VV((HH)| )| (H)(H)(H) (H) ((HH)()(--k) k) (G)((G)(--
k).k).
(k+1) s(k+1) sk+1 k+1 (G) ((G) (-k) s-k) skk
(k+1)(k+1) ssk+1k+1 (G) ((G) (--kk)) sskk andand
ssk+1 k+1 sskk is true whileis true while k+1 k+1 (G)((G)(--
k),k), i.e.,i.e., for for k k ((--1)1) / / ((++1)1)..
ProofProof
II(HH) = = 11++148148 xx + +147 147 xx22 + + 343343 xx3
is not unimodalis not unimodal
I(G) = 1 + 5x + 4x2 + x3 is log-concavelog-concave
G
K127K7
K7
K7
H = K127+3K7
Examples
GG andand HH are are perfectperfect
IfIf GG is ais a minimalminimal imperfect graphimperfect graph, then, then
II(G)(G) is log-concave.is log-concave.
I(C7) = 1 + 7x + 14 x2 + 7x3
C7
Remark
Example
There is anThere is an imperfectimperfect graphgraph GG whosewhose II((GG)) isis notnot
unimodal.unimodal.Example
G = K97+ 4K3~ C5
K97GGK3 K3
K3 K3
C5
I(GG) = 1 + 114x + 603x2 + 921x3 + 891x4 + 945x5 + 405x6
Remark
IfIf GG is a bipartite graph is a bipartite graph with with (G) = (G) = , then , then ssp p ssp+1 p+1 …… ss-1 -1 ss
wherewhere pp = = (2(2-1)/3-1)/3..
I(GG) = 1+8x+19x2
+20x3+10x4+2x5 G
= 5 ; p = 3
Corollary
Example
IfIf TT is a tree withis a tree with (T) = (T) = , thenthen
ssp p s sp+1 p+1 …… s s-1 -1 s s
wherewhere pp = = (2(2-1)/3-1)/3..
= 6 p = 4
I(TT) = 1 + 8x + 21x2
+26x3 +17x4 + 6x5 + x6 T
Corollary
Example
For P4 p=1
We found out that (sk) is decreasing in this upper part:
Conjecture 1: Conjecture 1: I(T)I(T) is unimodal for a is unimodal for a
treetree T. T.
1 2 3 2-1 3
k
sk
decreasing
Unimodal ? Log-concave ?
ifif TT is ais a treetree,then ,then ssp p s sp+1 p+1 …… s s-1 -1 s s , p = = (2(2(T)-(T)-
1)/31)/3..
V. E. Levit, E. Mandrescu, Graph Theory in Paris: Proceedings of a Conference in Memory of C. Berge (2006)
IntroductionIntroduction
Quasi-Regularizable GraphsQuasi-Regularizable Graphs
König-Egerváry GraphsKönig-Egerváry Graphs
The Main InequalityThe Main Inequality
Well-Covered GraphsWell-Covered Graphs
Perfect GraphsPerfect Graphs
Corona GraphsCorona Graphs
Palindromic GraphsPalindromic Graphs
G is called a König-Egerváry (K-E) graph if (G) + (G) = |
V(G)|. R. W. Deming, Discrete Mathematics 27 (1979)
If G is bipartite, then G is a König-Egerváry graph.
F. Sterboul, J. of Combinatorial Theory B 27 (1979)
Well-known !
(G) + (G) = 5
G
(H) + (H) < 6
H
If G is a König-Egerváry graph, then
Theorem
(i) sk tk, k = (G), where = (G) and
(ii) the coefficients sk satisfy
;
2
...
32
22
12 3
3
2
21
ssss
(1+2x) (1+x) = t0 + t1x +…+ t…+ t-1-1 x1+ t+ tx
(iii) sp ≥ sp+1 ≥… ≥ ss-1-1 ≥≥ s s for p = (21)/3 .
V. E. Levit, E. Mandrescu, Congressus Numerantium 179 (2006)
G H
2)()(
4)()(
HG
HG
Example
Proof
kts k
kk
2
k
k
k xk
x
0
2)21(
k
kk xsxGI
0
);(
k
kk xtxxxHI
0
)21()1();(
Proof
&
I(G) = s0 + s1x +…+ s…+ s-1-1 x1+ s+ sx
ks k
k
2
121
kk s
ks
k
1
1
1
1
1
:&
k
k
k
k
k
s
k
sRyanFisher
kk sksk 21 1
12
1
1
k
s
k
s
k
s k
k
k
kk
.3/)12(.,.
),(21)(21
kforei
kkforholdssks kk
If G has sk stable sets of size k, 1 k (G) = ,
then
Theorem
D. C. Fisher, J. Ryan, Discrete Mathematics 103 (1992)
L. Petingi, J. Rodriguez, Congressus Numerantium 146 (2000)
… and an alternative proof was given by
....
321
1
3
1
3
2
1
2
1
1
1
ssss
We found out that the sequence (sk) is
decreasing in this upper part:If a König-Egerváry graph G has (G) = ,
then
1 2 3 p k
sk
decreasing
Unimodal ? Log-concave ?
Unconstrained ?
sp sp+1 … s-1 s for p = (21)/3
ExampExamplele
G is a K-E
graph
I(G) = 1 + 13 x + 21 x2 + 35 x3
+35 x4+ 21 x5 + 7 x6 + 1 x7
p = (2-1)/3 = (14-1)/3 = 5
(G) = 7 (G) = 6 n = 13
21211335 < 0 I(G) is not log-concave, but
unimodal!
G = K6 + 7K1
s5 = 21 s6 = 7 s7 = 1
G K6
K1
K1K1
K1 K1
K1
K1
I(G) = 1 + 8x + 20x2 +23x3 +20x4 +1x5
unimodal Example
= 5, = 3,
p = (2-1)/3 = 3
I(G) is unimodal for every König-Egerváry graph G.
G
Conjecture 5
IntroductionIntroduction
Quasi-Regularizable GraphsQuasi-Regularizable Graphs
König-EgervKönig-Egervááry Graphsry Graphs
The Main InequalityThe Main Inequality
Well-Covered GraphsWell-Covered Graphs
Perfect GraphsPerfect Graphs
Corona GraphsCorona Graphs
Palindromic GraphsPalindromic Graphs
Recall : “Corona” Recall : “Corona” operation operation
P3
P4
K1 2K1
K3
G = PG = P44 {P {P3 3 , K, K1 1 , 2K, 2K1 1 , K, K33}}
Particular case of Particular case of “Corona”“Corona”K1
P4
K1K1 K1
G = PG = P4 4 K K 11
Each stable set of Each stable set of G = H G = H K K11 can can be enlarged to a maximum be enlarged to a maximum
stable set.stable set.
RemarkRemark
G is called well-covered if all its maximal stable sets are of the same size (M.D.
Plummer, 1970).
Def.Def.Equivalently, G is well-covered if each of its
stable sets is contained in a maximum stable set.
Let G be a graph of girth > 5, which is isomorphic to neither C7 nor K1. Then G is well–covered if and
only if G = H* for some graph H.
Theorem
A. Finbow, B. Hartnell, R. Nowakowski, J. Comb. Th B 57 (1993)
Appending a single pendant edge
to each vertex of H H*.
H* is very well-covered, for any graph H
Remark
If G is a graph of order n, and
I(G) = s0 + s1x +…+ s…+ s-1-1 x1+ s+ sx , then
and the formulae connecting the coefficients of I(G) and of I(G*) are:
)(0,)1(0
Gkkn
jnts
k
jj
jkk
nkkn
jnst
k
jjk
0,0
kGG
k
kk
Gn
knkG
kk
n
j
jj
xxsx
xxsxtxGI
)()(
0
)(
)(
00
*
)1()1(
)1(;
Theorem
V. E. Levit, E. Mandrescu, Discrete Applied Mathematics (2008) V. E. Levit, E. Mandrescu, Discrete Applied Mathematics (2008)
Well-covered spidersWell-covered spiders: :
Sn
A spider is a tree having at most one vertex of degree
> 2.
K2
K1
P4
Let T* be the tree obtained from the tree T by appending a
single pendant edge to each vertex of T.
T
((T**)) = the order ofthe order of T
( )*
ExamplExamplee
RemarkRemark
(T*) = 4
(iv) T is a is well-covered spider or T is
obtained from a well-covered tree T1 and a
well-covered spider T2, by adding an edge
joining two non-pendant vertices of T1,
T2, respectively.
For a tree T K1 the following are equivalent: (i) T is well-covered
(iii) T = L* for some tree L
Theorem
Appending a single pendant edge
to each vertex of H H*.
G. Ravindra, Well-covered graphs, J. Combin. Inform. System Sci. 2 (1977)
V. E. Levit, E. Mandrescu, Congressus Numerantium 139 (1999) V. E. Levit, E. Mandrescu, Congressus Numerantium 139 (1999)
(ii) T is very well-covered
the sequence (sk) is unconstrained with respect to
order in this upper part!
1 2 3 2-1 3
k
sk
increasing
unconstrained
2
decreasing
For every well-covered tree T, with (T) = ,
The independence polynomial The independence polynomial of anyof any wewell-coll-covvered spiderered spider
SSn n , n, n 1, 1, is unimodal andis unimodal and
mode(Sn) = n- (n-1)/3
Proposition
all are unimodal !
I(K1)=1+x
I(P4) = 1+4x+3x2
I(K2) = 1+2x
V. E. Levit, E. Mandrescu, Congresus Numerantium 159 (2002)V. E. Levit, E. Mandrescu, Congresus Numerantium 159 (2002)
The independence The independence polynomial of anypolynomial of any well–well–
covered spidercovered spider S Snn is log–is log–concave.concave.
Proof & “If P, Q are log-
concave, then PQ is log-concave.”
Proposition
V. E. Levit, E. Mandrescu, Carpathian J. of Math. 20 (2004)
Moreover,
n
k
kkn x
k
n
k
nxxSI
1 1
121)1();(
IntroductionIntroduction
Quasi-Regularizable GraphsQuasi-Regularizable Graphs
König-EgervKönig-Egervááry Graphsry Graphs
The Main InequalityThe Main Inequality
Well-Covered GraphsWell-Covered Graphs
Perfect GraphsPerfect Graphs
Corona GraphsCorona Graphs
Palindromic GraphsPalindromic Graphs
A (A (graphgraph) polynomial) polynomial
PP((xx)) = = aa0 0 + + aa11xx ++……++ aannxxnn is calledis called
palindromicpalindromic ifif aai i = a = an-i n-i , i = 0,1,..., , i = 0,1,..., n/2n/2..
P(x) = (1 + x)n
J. J. Kennedy J. J. Kennedy –– ““Palindromic graphsPalindromic graphs”” Graph Theory Notes of New York, XXII (1992)Graph Theory Notes of New York, XXII (1992)
nK1
v1 v2 v3 vn
In fact, (1+x)n = I(nK1)
DefinitionDefinition characteristicmatchingindependence
I. Gutman, I. Gutman, Independent vertex palindromic graphsIndependent vertex palindromic graphs, , Graph Theory Notes of New York, XXIII (1992)Graph Theory Notes of New York, XXIII (1992)
ExamplExamplee
(i) |S| q|NG(S)| for every stable setfor every stable set S ofof G;
Theorem
(ii) q(k+1)sk+1 (q+1)(-k)sk, 0 0 k k
< <
(iii) sr … s-1 s , r = r = ((q+1)((q+1) - q)/(2q+1) - q)/(2q+1) (iv) ifif q = 2, then then I(G) is palindromic is palindromic andand
LetLet G = HqK1 havehave (G) = andand (sk) be thebe the
coefficients ofcoefficients of I(G). Then the following are true:. Then the following are true:
s0 s1 … sp , p = p = (2(2+2)/5+2)/5
sr … s-1 s , r = r = (3(3-2)/5-2)/5 . .
We found out that the sequence (sk) is decreasing in this upper part:
ifif G = G = HqK1 has has (G) = , then, then
1 2 3 r k
sk
decreasing
Unimodal ? Log-concave ?
Unconstrained ?
sr … s-1 s , r = r = ((q+1)((q+1)-q)/(2q+1)-q)/(2q+1)
IfIf G = G = H2K1 , then , then I(GG) is palindromic and its sequence (sk) is increasing in its first part
and decreasing in its upper part !
1 2 3 3-2 5
k
sk
increasing
Unimodal ?
2+2 5
decreasing
Question:
Is I(GG) unimodal ?
K1,3 K1,3 = the “claw”
I(K1,3) = 1+4x+3x2+x3
is not palindromic.I(G) = 1+s1x+s2x2 = 1+nx+x2
1. If (G) = 2 and I(G) is palindromic, then n2, I(G) = 1 + n x + 1x2 and I(G) is log-concave,
and hence unimodal, as well.
Remarks
G = Kn–e, n2
2. If (G) = 3 and I(G) is palindromic, then n3, I(G) = 1 + n x + nx2 + 1x3 and I(G) is log-concave,
and hence unimodal, as well.
ExampleExampless
(G) = 5
G = K1832 + 4K7 + (K2K539) + 5K1
II(G) (G) = 1= 1++24062406xx++13821382xx22++13821382xx33++24062406xx44++11xx55
K1832
5K1
K2K539
4K7
s2 = 10+2539+677 =
1382
s4 = 5+7777 = 2406
s3 = 10+4777
= 1382
s1 = 5+28+1832+539+2 =
2406
IfIf GG has a stable sethas a stable set SS with:with: |N(A)|N(A)S| = 2|A|S| = 2|A| for every stable setfor every stable set
A A V(G) V(G) –– S S, , then then II((GG)) isis palindromicpalindromic..D. Stevanovic, D. Stevanovic, Graphs with palindromic independence polynomialGraphs with palindromic independence polynomial
Graph Theory Notes of New York XXXIV Graph Theory Notes of New York XXXIV (1998)(1998)
Theorem
S = { } II(G) (G) = 1+ = 1+ 55x x + + 55xx2 2 + 1+ 1xx33
ExamplExamplee
G
The condition that: “The condition that: “GG has a stable has a stable setset SS with:with: |N(A)|N(A)S| = 2|A|S| = 2|A| for every for every
stable stable set set A A V(G) V(G) –– S S”” isis NOTNOT necessarynecessary!!
Remark
G S = { }
II(G) (G) = 1+= 1+66xx++66xx22+1+1xx33
I. Gutman, I. Gutman, Independent vertex palindromic graphs,Independent vertex palindromic graphs, Graph Theory Notes of New York XXIII (1992) Graph Theory Notes of New York XXIII (1992)
ExamplExamplee
IfIf G = (V,E)G = (V,E) has has ss=1,s=1,s-1-1=|=|VV|| and the and the unique maximum stable setunique maximum stable set SS satisfies: satisfies: ||N(u)N(u)S| = 2S| = 2 for everyfor every uuV-SV-S,, then I(G) isis
palindromic.
Corollary
GS={ } II(G) (G) = 1 + = 1 + 99x x + + 2727xx2 2 + + 3838xx33+ +
+ 1+ 1xx6 6 + + 99xx5 5 + + 2727xx4 4
D. Stevanovic, D. Stevanovic, Graphs with palindromic independence polynomialGraphs with palindromic independence polynomial Graph Theory Notes of New York XXXIV Graph Theory Notes of New York XXXIV
(1998)(1998)
ExamplExamplee
RULE 1:RULE 1: If If is a is a clique coverclique cover of of GG, then: , then: for each clique for each clique CC,, addadd two new non- two new non-
adjacent vertices adjacent vertices andand join them to all the join them to all the vertices of vertices of CC.. The new graph is The new graph is
denoted bydenoted by {G}.{G}.
A A clique coverclique cover of of GG is a spanning graph of is a spanning graph of GG, , each component of which is a each component of which is a cliqueclique..
The set The set SS = { = {all these new verticesall these new vertices} is the unique } is the unique maximum stable set in the new graph maximum stable set in the new graph H = H = {G}{G}
and satisfies: and satisfies: |N(u)|N(u)S| = 2S| = 2 for any for any uuV(V(HH)-S)-S..Hence, Hence, II((HH)) isis palindromic by Stevanovic’s palindromic by Stevanovic’s
TheoremTheorem..
D. Stevanovic, Graphs with palindromic independence polynomial D. Stevanovic, Graphs with palindromic independence polynomial Graph Theory Notes of New York XXXIV Graph Theory Notes of New York XXXIV
(1998)(1998)
How to build graphs with palindromic independence polynomials ?
S={ }
|N(u)|N(u)S| = 2S| = 2, for any , for any uuV(H)-SV(H)-S
G
H = {G}
II(G) (G) = = 11++66xx++99xx22++22xx33
II(H) (H) == 11++1212xx++4848xx22++7676xx33++4848xx4 4 ++1212xx55++11xx66
= { }
ExamplExamplee
In particular:In particular: If If each cliqueeach clique of the clique of the clique cover cover of of GG consists of a consists of a single vertexsingle vertex, ,
then: the new graph then: the new graph {G}{G} is denoted by is denoted by GG○○2K2K11 . .
GG○○mKmK11 is theis the coronacorona ofof GG andand
mKmK11..
G○2K1
G
= { } II(G○2K(G○2K1) ) = 1 + = 1 + 1212x x + + 5353xx2 2 + + 120120xx33+ +
+ +156156xx44+1+1xx88+ + 1212xx7 7 + + 5353xx6 6 + + 120120xx55
ExamplExamplee
RULE 2.RULE 2. If If is a is a cycle covercycle cover of of GG, then:, then:(1)(1) add two pendant neighbors to add two pendant neighbors to each vertexeach vertex from from ;;(2) for (2) for each edge abeach edge ab of of , add two new vertices and join , add two new vertices and join them them to to aa & & bb;;(3) for (3) for each edge xyeach edge xy of a of a proper cycleproper cycle of of , add a new , add a new vertex vertex and join it to and join it to xx & & yy..
A A cycle covercycle cover of of GG is a spanning graph of is a spanning graph of GG, each , each component of which is a component of which is a vertexvertex, an , an edgeedge, or a , or a proper proper
cyclecycle..
The set The set SS = { = {ALL THESE NEW VERTICESALL THESE NEW VERTICES} is stable in the } is stable in the new graph new graph H = H = {G}{G} and satisfies: and satisfies: |N(v)|N(v)S| = 2 for any S| = 2 for any
vvV(V(HH)-S)-S.. Therefore,Therefore, II((HH)) isis palindromicpalindromic..
The new graph is denoted byThe new graph is denoted by {G}.{G}.
D. Stevanovic, Graphs with palindromic independence polynomial D. Stevanovic, Graphs with palindromic independence polynomial Graph Theory Notes of New York XXXIV Graph Theory Notes of New York XXXIV
(1998)(1998)
How to build graphs with palindromic independence polynomials ?
S={ }
|N(u)|N(u)S| = 2S| = 2, for any , for any uuV(H)-SV(H)-S
II(G) (G) = = 11++77xx++1313xx22++55xx33
II(H) (H) == 11++1515xx++8383xx22++218218xx33++298298xx44+218+218xx55++8383xx66++1515xx77++11xx88
= { } is a cycle cover
G
H = {G}
ExamplExamplee
Proposition
LetLet G = H2K1 havehave (G) = andand
(sk) be the coefficients ofbe the coefficients of I(G).
s0 s1 … sp , p = p =
(2(2+2)/5+2)/5 sr … s-1 s , r = r =
(3(3-2)/5-2)/5 . .
ThenThen I(G) is palindromic is palindromic andand
V. E. Levit, E. Mandrescu, 39th Southeastern Intl. Conf. on Combininatorics, Graph Theory, and Computing,
Florida Atlantic University, March 3-7, 2008
IfIf G = G = H2K1 , then , then I(GG) is palindromic and its sequence (sk) is increasing in its first part and decreasing in its upper part !
1 2 3 3-2 5
k
sk
increasing
Unimodal ?
2+2 5
decreasing
Question:
Is I(GG) unimodal ?
I(G) = 1 + 12x + 55x2 + 128x3 + 168x4 + 128x5 + 55x6 + 12x7 + x8
I(P4) == 1+1+44x++33x22
s0 = 1 s1 = 12 s2 = 55 s3 = 128 (p
= 3)
p = (2(G)+2)/5 = 3, r = (3(G)-3)/5 = 5
(G) = 8
s5 = 128 s6 = 55 s7 = 12 s8 = 1 (r
= 5)
ExamplExamplee
G = P4o2K1
P4
Theorem I(Pn2K1) has only real roots, and consequently is log-concave. Zhu Zhi-Feng, Australasian Journal of Combinatorics 38 (2007)
IfIf GG is is quasi-regularizablequasi-regularizable of order of order 22(G), , then then ssp p s sp+1 p+1 …… s s-1 -1 s s ,, pp = = (2(2--
1)/31)/3..
Theorem
ExampExamplele
G K6
K1K1
K1K1
K1
K1(G) = 6 n = 12
G is quasi-regularizableis quasi-regularizable
pp = = (12-1)/3(12-1)/3 = 4 = 4
ss4 4 = 15= 15 s s5 5 = 6= 6 s s6 6 = 1= 1
I(G) = 1 + 12 x + 15 x2 + 20 x3 + 15 x4 + 6 x5 + 1 x6
V. E. Levit, E. Mandrescu, Graph Theory in Paris: Proceedings of a Conference in Memory of C. Berge (2006)
Theorem 2
(s1)2 s0 s2 , (s2)2 s1 s3 andand
(s-1)2 s s-2 , (s-2)2 s-1 s-3
I(G) is palindromic, while its coefficients is palindromic, while its coefficients (sk) satisfy: satisfy:
If If is a cycle cover of is a cycle cover of H without without vertex-cyclesvertex-cycles and and G = {H}
has has (G) = , then , then G is quasi-regularizable of order is quasi-regularizable of order 2 and and
s0 s1 … sp , p = (+1)/3 sq … s-1 s , q = (2-1)/3
V. E. Levit, E. Mandrescu, 39th Southeastern Intl. Conf. on Combinatorics, Graph Theory, and Computing, Florida Atlantic University, March 3-7, 2008
(i)
(ii)
1 2 3 2-1 3
k
sk
increasing
Unimodal ?
+1 3
decreasing
If is a cycle cover of H without vertex-vertex-
cyclescycles, , G = {H} has (G) = , then I(G) is palindromic and its sequence (sk) is increasing in its first part and decreasing in its upper part !
Question: Is I(G) unimodal ?
G = {H}
II(H) (H) = = 11++88xx++1919xx22++1313xx33++xx44
II(G) (G) == 11++1616xx++9595xx22++265265xx33++371371xx44+265+265xx55++9595xx66++1616xx77++11xx88
S={ }
= { } H
s0 s1 s2 s3 , p = (+1)/3 = 3
s5 s6 s7 s8 , q = (2-1)/3 = 5
(G) = 8
Example
Trees
Bipartite
König-Egervár
y graphs
Perfect
Very well-
covered
well-covere
d
quasi-regularizab
le
GG○○qKqK11
GG○○2K2K11 & GG perfect
GG○○KK11
GG○○KKpp
{G} & is a cycle cover of
G
Some family relationships
TT○○2K2K11 &
T =T = a tree
==
Problem 1Problem 1
Find an inequality leading Find an inequality leading to partial log-concavity of to partial log-concavity of
the independence the independence polynomial.polynomial.
For very-well covered graphs: (S(S-1-1))22 S S S S-2-2
ExampleExample
V. E. Levit, E. Mandrescu, Graph Theory in Paris: Proceedings of a Conference in Memory of C. Berge (2006)
P(x) = (1 + x)n
Problem 22Characterize polynomials Characterize polynomials that that areare independence independence
polynomials.polynomials.
P(x) = I(nK1)
but, there is no graph G whose I(G) = 1 + 4x + 17x2
C. Hoede, X. Li C. Hoede, X. Li Discrete Mathematics 125 (1994)Discrete Mathematics 125 (1994)
Example
Problem 3Problem 3
Characterize the graphs whose Characterize the graphs whose independence polynomialsindependence polynomials
are are palindromicpalindromic.. D. Stevanovic D. Stevanovic
Graph Theory Notes of New York XXXIV (1998)Graph Theory Notes of New York XXXIV (1998)A graph A graph GG with with (G) = 2(G) = 2 has has
a palindromic a palindromic independence polynomialindependence polynomial
iff iff G = KG = Knn- e- e.. I(G) == 11 ++ nn x ++ 11 x2 (G) = 2
Example