Maximum Multiplicity of a Root of the Matching Polynomial of a Tree and Minimum Path Cover

TL;DR

This paper establishes a precise condition linking the maximum multiplicity of a root of a tree's matching polynomial to its minimum path cover, advancing understanding of graph polynomial roots.

Contribution

It provides a necessary and sufficient condition connecting root multiplicity and minimum path cover in trees, a novel theoretical insight.

Findings

Characterizes maximum root multiplicity in trees

Links root multiplicity to minimum path cover

Advances theoretical understanding of matching polynomials

Abstract

We give a necessary and sufficient condition for the maximum multiplicity of a root of the matching polynomial of a tree to be equal to the minimum number of vertex disjoint paths needed to cover it.