The 3-connectivity of a graph and the multiplicity of zero "2" of its chromatic polynomial

Abstract

Let G be a graph of order n, maximum degree δ, and minimum degree δ. Let P(G, λ) be the chromatic polynomial of G. It is known that the multiplicity of zero "0" of P(G, λ) is one if G is connected, and the multiplicity of zero "1" of P(G, λ) is one if G is 2-connected. Is the multiplicity of zero "2" of P(G, λ) at most one if G is 3-connected? In this article, we first construct an infinite family of 3-connected graphs G such that the multiplicity of zero "2" of P(G, λ) is more than one, and then characterize 3-connected graphs G with δ + δ≥n such that the multiplicity of zero "2" of P(G, λ) is at most one. In particular, we show that for a 3-connected graph G, if δ + δ≥n and (δ, δ 3)≠(n-3, 3), where δ 3 is the third minimum degree of G, then the multiplicity of zero "2" of P(G, λ) is at most one. © 2011 Wiley Periodicals, Inc.