On the roots of hypergraph chromatic polynomials

TL;DR

This paper extends bounds on the roots of chromatic polynomials from graphs to uniform hypergraphs using the theory of bounded exponential type graph polynomials.

Contribution

It generalizes Sokal's root bounds for graph chromatic polynomials to the setting of uniform hypergraphs.

Findings

Roots of hypergraph chromatic polynomials are bounded in absolute value by a constant times the maximum degree.

Uses the theory of bounded exponential type graph polynomials to establish the bounds.

Provides a framework for analyzing roots of hypergraph polynomials.

Abstract

Let $G = (V,E)$ be a finite, simple, connected graph with chromatic polynomial $P_G(q)$. Sokal \cite{sokal} proved that the roots of the chromatic polynomial of $G$ are bounded in absolute value by $KD$ where, $D$ is the maximum degree of the graph and $7< K < 8$ is a constant. In this paper we generalize this result to uniform hypergraphs. To prove our results we will use the theory of the bounded exponential type graph polynomials.