On the Chromatic Roots of Generalized Theta Graphs

The generalized theta graph ?s1, ?, sk consists of a pair of endvertices joined by k internally disjoint paths of lengths s1, ?, sk?1. We prove that the roots of the chromatic polynomial ?(?s1, ?, sk, z) of a k-ary generalized theta graph all lie in the disc|z?1|?1+o(1)]k/logk, uniformly in the path lengths si. Moreover, we prove that ?2, ?, 2?K2, k indeed has a chromatic root of modulus 1+o(1)]k/logk. Finally, for k?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.