Cutpoints and the chromatic polynomial
Earl Glen Whitehead, Lian‐Chang Zhao · Journal of Graph Theory · 1984
Abstract We prove that the multiplicity of the root 1 in the chromatic polynomial of a simple graph G is equal to the number of nontrivial blocks in G. In particular, a connected simple graph G has a cutpoint if and only if its chromatic polynomial is divisible by (λ – 1)2. We apply this theorem to obtain some chromatic equivalence and uniqueness results.