A refined Gallai-Edmonds structure theorem for weighted matching polynomials

TL;DR

This paper refines the Gallai-Edmonds structure theorem for weighted matching polynomials, linking it to continued fractions and classical polynomial zero theorems, and explores properties of these zeros.

Contribution

It introduces a refined theorem connecting weighted matching polynomials with branched continued fractions and classical polynomial zero results.

Findings

Refined Gallai-Edmonds structure theorem for weighted matching polynomials

Connection between matching polynomials and branched continued fractions

Results on zeros of matching polynomials

Abstract

In this work, we prove a refinement of the Gallai-Edmonds structure theorem for weighted matching polynomials by Ku and Wong. Our proof uses a connection between matching polynomials and branched continued fractions. We also show how this is related to a modification by Sylvester of the classical Sturm's theorem on the number of zeros of a real polynomial in an interval. In addition, we obtain some other results about zeros of matching polynomials.