On the minimum real roots of the adjoint polynomial of a graph

Zhao, Haixing; Liu, Ruying

doi:10.1016/j.disc.2009.01.015

Cited by 4 publications

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“…The adjoint polynomial was introduced by R. Liu [8] and it is studied in a series of papers ( [1,2,11,10,12]). Let us remark that we are using the definition for the adjoint polynomial from [4], but usually it is defined without alternating signs.…”

Section: Introductionmentioning

confidence: 99%

One more remark on the adjoint polynomial

Bencs

2017

European Journal of Combinatorics

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Abstract. The adjoint polynomial of G iswhere a k (G) denotes the number of ways one can cover all vertices of the graph G by exactly k disjoint cliques of G. In this paper we show the the adjoint polynomial of a graph G is a simple transformation of the independence polynomial of another graph G. This enables us to use the rich theory of independence polynomials to study the adjoint polynomials. In particular we a give new proofs of several theorems of R. Liu and P. Csikvári.

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Section: Introductionmentioning

confidence: 99%

One more remark on the adjoint polynomial

Bencs

2017

European Journal of Combinatorics

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“…The partition polynomial studied by Wagner reduces to a σ‐polynomial and the σ‐polynomial of the complement of a triangle free graph is equal to the matching generating polynomial under a simple transformation . Moreover, the widely studied adjoint polynomial (see, for example, [, , , , , , ]) is equal to the σ‐polynomial of the complement of the graph. The authors in investigate the rook and chromatic polynomials, and prove that every rook vector is a chromatic vector.…”

Section: Introductionmentioning

confidence: 99%

On the Roots of σ‐Polynomials

Brown

Erey

2015

Journal of Graph Theory

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Given a graph G of order n, the σ‐polynomial of G is the generating function σ(G,x)=∑aixi where ai is the number of partitions of the vertex set of G into i nonempty independent sets. Such polynomials arise in a natural way from chromatic polynomials. Brenti (Trans Am Math Soc 332 (1992), 729–756) proved that σ‐polynomials of graphs with chromatic number at least n−2 had all real roots, and conjectured the same held for chromatic number n−3. We affirm this conjecture.

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Two remarks on the adjoint polynomial

Csíkvári

2012

European Journal of Combinatorics

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On the minimum real roots of the adjoint polynomial of a graph

Cited by 4 publications

References 14 publications

One more remark on the adjoint polynomial

One more remark on the adjoint polynomial

On the Roots of σ‐Polynomials

Two remarks on the adjoint polynomial

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