On the chromatic roots of generalized theta graphs

The generalized theta graph Theta (S1,....Sk) consists of a pair of endvertices joined by k internally disjoint paths of lengths s(1),..., s(k) greater than or equal to 1. We prove that the roots of the chromatic polynomial pi(Theta (S1,...,Sk), z) of a k-ary generalized theta graph all lie in the disc \z- 1 \ less than or equal to [1 +o(1)] k/log k, uniformly in the path lengths s(1). Moreover, we prove that Theta (2,...,2) similar or equal to K-2,K-k indeed has a chromatic root of modulus [1 + o(1)] k/log k. Finally, for k less than or equal to 8 we prove that the generalized theta graph with a chromatic root that maximizes \ z - 1 \ is the one with all path lengths equal to 2; we conjecture that this holds for all k. (C) 2001 Academic Press.