New upper bounds for the greatest number of proper colorings of a (V,E)-graph

Let ?? denote the family of simple undirected graphs on v vertices having e edges ((v, e)-graphs) and P(G; λ) be the chromatic polynomial of a graph G. For the given integers v, e, and λ, let f(v, e, λ) denote the greatest number of proper colorings in λ or less colors that a (v, e)-graph G can have, i.e., f(v, e, λ) = max{P(G; λ): G ∈ ??}. In this paper we determine some new upper bounds for f(v, e, λ).