Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions
Combinatorics, Probability and Computing
Matching Polynomials And Duality
Combinatorica
Absence of Zeros for the Chromatic Polynomial on Bounded Degree Graphs
Combinatorics, Probability and Computing
Analytic Combinatorics
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One can define the adjoint polynomial of the graph G as follows. Let a"k(G) denote the number of ways one can cover all vertices of the graph G by exactly k disjoint cliques of G. Then the adjoint polynomial of G is h(G,x)=@?k=1n(-1)^n^-^ka"k(G)x^k, where n denotes the number of vertices of the graph G. In this paper we show that the largest real root @c(G) of h(G,x) has the largest absolute value among the roots. We also prove that @c(G)@?4(@D-1), where @D denotes the largest degree of the graph G. This bound is sharp.