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

Let G be a graph of order n, maximum degree ?, and minimum degree d. Let P(G, ?) be the chromatic polynomial of G. It is known that the multiplicity of zero 0 of P(G, ?) is one if G is connected, and the multiplicity of zero 1 of P(G, ?) is one if G is 2-connected. Is the multiplicity of zero 2 of P(G, ?) 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, ?) is more than one, and then characterize 3-connected graphs G with ? + d?n such that the multiplicity of zero 2 of P(G, ?) is at most one. In particular, we show that for a 3-connected graph G, if ? + d?n and (?, d3)?(n-3, 3), where d3 is the third minimum degree of G, then the multiplicity of zero 2 of P(G, ?) is at most one. (c) 2011 Wiley Periodicals, Inc. J Graph Theory