Algebraic invariants arising from the chromatic polynomials of theta graphs

Abstract

This paper investigates some algebraic properties of the chromatic polynomials of theta graphs, i.e. graphs which have three internally disjoint paths sharing the same two distinct end vertices. We give a complete description of the Galois group, discriminant and ramification indices for the chromatic polynomials of theta graphs with three consecutive path lengths. We then do the same for theta graphs with three paths of the same length, by comparing them algebraically to the first family. This algebraic link extends naturally to generalised theta graphs with k + 1 branches.