On the Roots of Chromatic Polynomials

Abstract

It is proved that the chromatic polynomial of a connected graph with n vertices and m edges has a root with modulus at least (m-1)/(n-2); this bound is best possible for trees and 2-trees (only). It is also proved that the chromatic polynomial of a graph with few triangles that is not a forest has a nonreal root and that there is a graph with n vertices whose chromatic polynomial has a root with imaginary part greater thann/4. © 1998 Academic Press.