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

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