Independence Polynomials and Hypergeometric Series

TL;DR

This paper characterizes chordal graphs as the only graphs whose independence polynomial inverse has a Horn hypergeometric power series expansion, linking graph theory with hypergeometric functions.

Contribution

It provides a novel characterization of chordal graphs through the hypergeometric nature of the inverse independence polynomial.

Findings

Chordal graphs are uniquely characterized by the hypergeometric expansion of the inverse independence polynomial.

The inverse of the independence polynomial for these graphs is Horn hypergeometric.

This links graph structure to special functions in a new way.

Abstract

Let $\Gamma$ be a simple graph and $I_\Gamma(x)$ its multivariate independence polynomial. The main result of this paper is the characterization of chordal graphs as the only $\Gamma$ for which the power series expansion of $I_\Gamma^{-1}(x)$ is Horn hypergeometric.