On the Chromatic Roots of Generalized Theta Graphs

The generalized theta graph s 1 ;:::;s k consists of a pair of endvertices joined by k internally disjoint paths of lengths s 1 ; : : :; s k 1. We prove that the roots of the chromatic polynomial ((s 1 ;:::;s k ; z) of a k-ary generalized theta graph all lie in the disc jz ? 1j 1 + o(1)] k= logk, uniformly in the path lengths s i. Moreover, we prove that 2;:::;2 ' K 2;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 jz ? 1j is the one with all path lengths equal to 2; we conjecture that this holds for all k.