The roots of σ-polynomials

Abstract

Let G be a connected graph. We denote by σ(G,x) and δ(G) respectively the σ-polynomial and the edge-density of G, where \(\Gamma ^{(m)} , \Gamma ^{(m)} (E) \cap \Gamma ^{(m)} (F)\). If σ(G,x) has at least an unreal root, then G is said to be a σ-unreal graph. Let δ(n) be the minimum edgedensity over all n vertices graphs with σ-unreal roots. In this paper, by using the theory of adjoint polynomials, a negative answer to a problem posed by Brenti et al. is given and the following results are obtained: For any positive integer a and rational number 0≤c≤1, there exists at least a graph sequence {Gi}1≤i≤a such that Gi is σ-unreal and δ(Gi)→c as n→∞ for all 1 ≤i≤a, and moreover, δ(n)→0 as n→∞.