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 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 paper, 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 ≥ 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 ≥ n and (∆, δ3) 6= (n− 3, 3), where δ3 is the third minimum degree of G, then the multiplicity of zero ‘2’ of P (G,λ) is at most one.