Independence polynomials of well-covered graphs: Generic counterexamples for the unimodality conjecture

Levit, Vadim E.; Mândrescu, Eugen

doi:10.1016/j.ejc.2005.04.007

Cited by 27 publications

(13 citation statements)

References 15 publications

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“…For the unimodality of independence polynomials of well-covered graphs, we refer readers to [6,21,22,23,28] for details.…”

Section: Theorem 22 For Two Vertex Disjoint Graphs Gmentioning

confidence: 99%

Unimodality of the independence polynomials of some composite graphs

Zhu

2017

Filomat

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Let I(G;x) denote the independence polynomial of a graph G. In this paper we study the unimodality properties of I(G; x) for some composite graphs G. Given two graphs G1 and G2, let G1[G2] denote the lexicographic product of G1 and G2. Assume I(G1; x) = P i_0 aixi and I(G2; x) = P i_0 bixi, where I(G2; x) is log-concave. Then we prove (i) if I(G1; x) is logconcave and (a2i ??ai??1ai+1)b21 _ aiai??1b2 for all 1 _ i _ _(G1), then I(G1[G2]; x) is log-concave; (ii) if ai??1 _ b1ai for 1 _ i _ _(G1), then I(G1[G2]; x) is unimodal. In particular, if ai is increasing in i, then I(G1[G2]; x) is unimodal. We also give two su_cient conditions when the independence polynomial of a complete multipartite graph is unimodal or log-concave. Finally, for every odd positive integer _ > 3, we find a connected graph G not a tree, such that _(G) = _, and I(G; x) is symmetric and has only real zeros. This answers a problem of Mandrescu and Miric?a.

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“…For the unimodality of independence polynomials of well-covered graphs, we refer readers to [6,21,22,23,28] for details.…”

Section: Theorem 22 For Two Vertex Disjoint Graphs Gmentioning

confidence: 99%

Unimodality of the independence polynomials of some composite graphs

Zhu

2017

Filomat

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“…They certainly form a sequence of positive integers. Alavi et al [1] showed, in general, that the independence sequence i k of a graph G can be far from unimodal, for example, the graph K 25 + 4K 2 has independence sequence 1, 33, 24,32,16 . More examples of graphs with nonunimodal independence sequences can be found in [1].…”

Section: Introductionmentioning

confidence: 99%

“…Finally, Levit and Mandrescu [24] amended the original unimodality conjecture on well-covered graphs as follows. A very well-covered graph G of order n (that is, on n vertices) is a well-covered graph for which every maximal independent set has size n/2; for example, the complete bipartite graphs K m,m are very well-covered.…”

Section: Introductionmentioning

confidence: 99%

On the unimodality of independence polynomials of very well-covered graphs

Brown

Cameron

2018

Discrete Mathematics

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The independence polynomial i(G, x) of a graph G is the generating function of the numbers of independent sets of each size. A graph of order n is very well-covered if every maximal independent set has size n/2. Levit and Mandrescu conjectured that the independence polynomial of every very well-covered graph is unimodal (that is, the sequence of coefficients is nondecreasing, then nonincreasing). In this article we show that every graph is embeddable as an induced subgraph of a very well-covered graph whose independence polynomial is unimodal, by considering the location of the roots of such polynomials. * Communicating author. Jason.Brown@dal.ca arXiv:1709.08236v1 [math.CO]

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“…Several results concerning the independence polynomials of well-covered graphs are presented in [4,11,12,13]. It is known that there exist well-covered graphs whose independence polynomials are not unimodal [16,24].…”

Section: Introductionmentioning

confidence: 99%