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Algebraic properties of chromatic roots
journal contributionposted on 2017-01-01, 00:00 authored by P J Cameron, Kerri Morgan
A chromatic root is a root of the chromatic polynomial of a graph. Any chromatic root is an algebraic integer. Much is known about the location of chromatic rootsin the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institutein late 2008. The purpose of this paper is to report on the seminar and subsequent developments.We conjecture that, for every algebraic integer alpha, there is a natural number n such that alpha+n is a chromatic root. This is proved for quadratic integers; an extension to cubic integers has been found by Adam Bohn. The idea is to consider certain special classes of graphs for which the chromatic polynomial is a product of linear factors and one interesting factor of larger degree. We also report computational results on the Galois groups of irreducible factors of the chromatic polynomial for some special graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.