Two remarks on the adjoint polynomial

Csíkvári, Péter

doi:10.1016/j.ejc.2011.11.010

Cited by 5 publications

(9 citation statements)

References 14 publications

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“…On the other hand, it can be proved that the sharp upper bound for the roots is 4(∆ − 1) in both cases. For the matching polynomial this is proved in [8], for the adjoint polynomial this is proved in [5].…”

Section: 2mentioning

confidence: 95%

Benjamini–Schramm continuity of root moments of graph polynomials

Csikvri

Frenkel

2016

European Journal of Combinatorics

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Article history:Available online xxxx a b s t r a c t Recently, M. Abért and T. Hubai studied the following problem. The chromatic measure of a finite simple graph is defined to be the uniform distribution on its chromatic roots. Abért and Hubai proved that for a Benjamini-Schramm convergent sequence of finite graphs, the chromatic measures converge in holomorphic moments. They also showed that the normalized logarithm of the chromatic polynomial converges to a harmonic real function outside a bounded disc.In this paper we generalize their work to a wide class of graph polynomials, namely, multiplicative graph polynomials of bounded exponential type. A special case of our results is that for any fixed complex number v 0 the measures arising from the Tutte polynomial Z Gn (z, v 0 ) converge in holomorphic moments if the sequence (G n ) of finite graphs is Benjamini-Schramm convergent.This answers a question of Abért and Hubai in the affirmative. Even in the original case of the chromatic polynomial, our proof is considerably simpler.

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Section: 2mentioning

confidence: 95%

Benjamini–Schramm continuity of root moments of graph polynomials

Csikvri

Frenkel

2016

European Journal of Combinatorics

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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%

“…For instance it has a real zero whose modulus is the largest among all zeros. H. Zhao showed ( [10]) that the adjoint polynomial always has a real zero, furthermore P. Csikvári proved ( [4]) that the largest real zero has the largest modulus among all zeros. He also showed that the absolute value of the largest real zero is at most 4(∆ − 1), where ∆ is the largest degree of the graph G.…”

Section: Introductionmentioning

confidence: 99%

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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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“…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

Cited by 5 publications

References 14 publications

Benjamini–Schramm continuity of root moments of graph polynomials

Benjamini–Schramm continuity of root moments of graph polynomials

One more remark on the adjoint polynomial

On the Roots of σ‐Polynomials

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