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On the chromatic roots of generalized theta graphs

Jason Brown·Carl Hickman·Alan D. Sokal·David G. Wagner
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Abstract

The generalized theta graph \Theta_{s_1,...,s_k} consists of a pair of endvertices joined by k internally disjoint paths of lengths s_1,...,s_k \ge 1. We prove that the roots of the chromatic polynomial $pi(\Theta_{s_1,...,s_k},z) of a k-ary generalized theta graph all lie in the disc |z-1| \le [1 + o(1)] k/\log k, uniformly in the path lengths s_i. Moreover, we prove that \Theta_{2,...,2} \simeq K_{2,k} indeed has a chromatic root of modulus [1 + o(1)] k/\log k. Finally, for k \le 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.

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Jason Brown, Carl Hickman, Alan D. Sokal, David G. Wagner. 2000-12-05. On the chromatic roots of generalized theta graphs. https://doi.org/10.1006/jctb.2001.2057

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