Inequalities Involving the Inequalities Involving the Coefficients of Independence Coefficients of Independence Polynomials Polynomials Combinatorial and Probabilistic Inequalities Isaac Newton Institute for Mathematical Sciences Cambridge, UK - June 23-27, 2008 Vadim E. Levit Vadim E. Levit 1,2 1,2 1 Ariel University Center of Samaria, Ariel University Center of Samaria, Israel Israel Eugen Mandrescu Eugen Mandrescu 2 2 2 2 Holon Institute of Technology, Holon Institute of Technology, Israel Israel
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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…
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
There are non-isomorphic graphs with I(G) = I(H)
G H
I(G) = I(H) = 1+6x+4x2
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
““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)
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)
What is known aboutWhat is known about II(T)(T),,
wherewhere TT is a treeis a tree??
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
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)
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
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
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..
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)
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,
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
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
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
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
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