An Analogue of the Gallai-Edmonds Structure Theorem for Nonzero Roots of the Matching Polynomial

TL;DR

This paper extends classical matching theory results to roots of the matching polynomial other than zero, providing structural insights and proving that vertex transitive graphs have simple roots.

Contribution

It establishes analogues of the Stability Lemma and Gallai-Edmonds Structure Theorem for arbitrary roots of the matching polynomial, generalizing classical matching theory.

Findings

Analogues of the Stability Lemma and Gallai's Lemma for any root of the matching polynomial.

The matching polynomial of a vertex transitive graph has simple roots.

Structural understanding of how graph matching properties change with vertex deletion.

Abstract

Godsil observed the simple fact that the multiplicity of 0 as a root of the matching polynomial of a graph coincides with the classical notion of deficiency. From this fact he asked to what extent classical results in matching theory generalize, replacing "deficiency" with multiplicity of $\theta$ as a root of the matching polynomial. We prove an analogue of the Stability Lemma for any given root, which describes how the matching structure of a graph changes upon deletion of a single vertex. An analogue of Gallai's Lemma follows. Together these two results imply an analogue of the Gallai-Edmonds Structure Theorem. Consequently, the matching polynomial of a vertex transitive graph has simple roots.