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Inequalities Involving the Coefficients of Independence Polynomials

I(G;x) = the independence polynomial of graph G. Results and conjectures on I(G;x) for some graph classes

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Inequalities Involving the Coefficients of Independence Polynomials

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    1. Inequalities Involving the Coefficients of Independence Polynomials

    4. A set of pairwise non-adjacent vertices is called a stable set or an independent set.

    5. If sk denotes the number of stable sets of size k in a graph G with ?(G) = ?, then I(G) = I(G;x) = s0 + s1x +…+ s?x? is called the independence polynomial of G.

    6. All the stable sets of G : …… ? …… {a}, {b}, {c}, {d}, {e}, {f} ….. {a, b}, {a, d}, {a, e}, {a, f}, {b, c}, {b, e}, {b, f}, {d, f}, …… { a, b, e }, { a, b, f }, { a, d, f }

    7. There are non-isomorphic graphs with I(G) = I(H)

    8. ALSO non-isomorphic trees can have the same independence polynomial !

    9. … for historical reasons, recall

    10. … for historical reasons

    13. “Clique polynomial”: C(G;x) = I(H;-x), where H is the complement of G, Goldwurm & Santini - 2000

    14. Add connections with Tutt polyunomial etc.Add connections with Tutt polyunomial etc.

    16. How to compute the independence polynomial ?

    18. How to compute the independence polynomial ?

    20. If G = (V,E), v?V and uv ?E, then the following assertions are true:

    23. A sequence of reals a0, a1,..., an is: (i) unimodal if a0 ? a1 ? ... ? am ? ... ? an for some m?{0,1,...,n}, (ii) log-concave if ak-1 ak+1 ? (ak)2 for every k ? {1,...,n-1}.

    24. A polynomial P (x) = a0 + a1x +…+ anxn is unimodal (log-concave) if its sequence of coefficients a0 , a1 , a2 , ... , an is unimodal (log-concave, respectively).

    27. For any permutation ? of the set {1, 2, …, ?}, there is a graph G such that ?(G) = ? and s?(1)< s?(2)< s?(3)< … < s?(?) where sk is the number of stable sets in G of size k.

    29. If all the roots of a polynomial with positive coefficients are real, then the polynomial is log-concave.

    30. What is known about I(T), where T is a tree?

    31. If T is a tree, then I(T) is unimodal.

    32. If F is a forest, then I(F) is unimodal.

    34. (i) log-concave ? unimodal = unimodal; Proof of (ii) using (i) and the converse of (i)! Proof of (ii) using (i) and the converse of (i)!

    35. The unimodality of independence polynomials of trees does not directly implies the unimodality of independence polynomials of forests !

    39. If G is a graph with ?(G) = ? and ?(G) = ?, then

    40. Let H = (A,B,E) be a bipartite graph with X?A ? X is a k-stable set in G (|X|=k) Y?B ? Y is a (k+1)-stable set in G, and XY?E ? X ? Y

    43. If G is a quasi-regularizable graph of order n = 2?(G) = 2?, then

    44. (i) If S is stable and |S| = k ? 2 |S| ? |S?N(S)| ? 2(?-k) = 2(?-|S|) ? n-|N[S]| ? ??-k ? 2 (?-k)

    50. A graph G is called well-covered if all its maximal stable sets are of the same size (namely, ?(G)).

    53. i.e., Conjecture 3 is true for every well-covered graph G having ?(G) ? 3. Check their result for ?(G) >7.Check their result for ?(G) >7.

    55. G = 4K10 + K4, 4, …, 4 Check their result for ?(G) >7. Be prepared to show that the polynomials for 1700 (and the smaller numbers) are unimodal, and in the interval [17**, infinity] they are also unimodal.Check their result for ?(G) >7. Be prepared to show that the polynomials for 1700 (and the smaller numbers) are unimodal, and in the interval [17**, infinity] they are also unimodal.

    57. Check their result for ?(G) >7. Be prepared to show that the polynomials for 1700 (and the smaller numbers) are unimodal, and in the interval [17**, infinity] they are also unimodal.Check their result for ?(G) >7. Be prepared to show that the polynomials for 1700 (and the smaller numbers) are unimodal, and in the interval [17**, infinity] they are also unimodal.

    58. Try general n instead of just 1701.Try general n instead of just 1701.

    59. Try general n instead of just 1701.Try general n instead of just 1701.

    60. Check their result for ?(G) >7. Be prepared to show that the polynomials for 1700 (and the smaller numbers) are unimodal, and in the interval [17**, infinity] they are also unimodal.Check their result for ?(G) >7. Be prepared to show that the polynomials for 1700 (and the smaller numbers) are unimodal, and in the interval [17**, infinity] they are also unimodal.

    61. Check their result for ?(G) >7. Be prepared to show that the polynomials for 1700 (and the smaller numbers) are unimodal, and in the interval [17**, infinity] they are also unimodal.Check their result for ?(G) >7. Be prepared to show that the polynomials for 1700 (and the smaller numbers) are unimodal, and in the interval [17**, infinity] they are also unimodal.

    63. If G is a well-covered graph with ?(G) = ?, then

    64. each (k+1)-stable set includes k+1 stable sets of size k (k+1) sk+1 Add the original proof!Add the original proof!

    65. If G is a very well-covered graph with ?(G) = ?, then

    66. (iv) Combining (ii) and (iii), it follows that I(G) is unimodal, whenever ? ? 9.

    73. If G is a perfect graph with ?(G) = ? and ?(G) = ?, then sp ? sp+1 ? … ? s?-1 ? s? where p = ?(??? ? 1) / (? ? 1)?.

    75. If S is stable and |S| = k, then H = G-N[S] has ?(H) ? ?(G)-k.

    77. If G is a minimal imperfect graph, then I(G) is log-concave.

    78. There is an imperfect graph G whose I(G) is not unimodal.

    79. If G is a bipartite graph with ?(G) = ?, then sp ? sp+1 ? … ? s?-1 ? s? where p = ?(2?-1)/3?.

    80. If T is a tree with ?(T) = ?, then sp ? sp+1 ? … ? s?-1 ? s? where p = ?(2?-1)/3?.

    83. G is called a König-Egerváry (K-E) graph if ?(G) + ?(G) = |V(G)|.

    84. If G is a König-Egerváry graph, then

    85. Add the original proof!Add the original proof!

    86. Add the original proof!Add the original proof!

    96. Well-covered spiders :

    97. Let T* be the tree obtained from the tree T by appending a single pendant edge to each vertex of T.

    100. The independence polynomial of any well-covered spider Sn , n ? 1, is unimodal and

    101. The independence polynomial of any well–covered spider Sn is log–concave.

    103. A (graph) polynomial P(x) = a0 + a1x +…+ anxn is called palindromic if ai = an-i , i = 0,1,..., ?n/2?.

    110. If G has a stable set S with: |N(A)?S| = 2|A| for every stable set A ? V(G) – S, then I(G) is palindromic.

    111. The condition that: “G has a stable set S with: |N(A)?S| = 2|A| for every stable set A ? V(G) – S” is NOT necessary!

    112. If G = (V,E) has s?=1,s?-1=|V| and the unique maximum stable set S satisfies: |N(u)?S| = 2 for every u?V-S, then I(G) is palindromic.

    113. RULE 1: If ? is a clique cover of G, then: for each clique C??, add two new non-adjacent vertices and join them to all the vertices of C. The new graph is denoted by ?{G}.

    114. |N(u)?S| = 2, for any u?V(H)-S

    116. RULE 2. If ? is a cycle cover of G, then: (1) add two pendant neighbors to each vertex from ?; (2) for each edge ab of ?, add two new vertices and join them to a & b; (3) for each edge xy of a proper cycle of ?, add a new vertex and join it to x & y.

    117. |N(u)?S| = 2, for any u?V(H)-S

    121. If G is quasi-regularizable of order 2?(G), then sp ? sp+1 ?…? s?-1 ? s? , p = ?(2?-1)/3?.

    130. … Thank you ! ... ???? !

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