Christoffel-Darboux type identities for independence polynomial

TL;DR

This paper introduces Christoffel-Darboux type identities for independence polynomials, providing new proofs and insights into properties like real roots of independence polynomials in claw-free graphs and related conjectures.

Contribution

It presents novel Christoffel-Darboux identities for independence polynomials and offers new proofs for known theorems and conjectures in graph theory.

Findings

Independence polynomial of claw-free graphs has only real roots

New Christoffel-Darboux identities for independence polynomials

Alternative proof of Chudnovsky and Seymour's theorem

Abstract

In this paper we introduce some Christoffel-Darboux type identities for independence polynomials. As an application, we give a new proof of a theorem of M. Chudnovsky and P. Seymour, claiming that the independence polynomial of a claw-free graph has only real roots. Another application is related to a conjecture of Merrifield and Simmons.