Graphs with Integer Matching Polynomial Roots

TL;DR

This paper investigates graphs with matching polynomials that have only integer roots, providing characterizations of such graphs, especially traceable, regular, and claw-free graphs, and exploring properties related to perfect matchings.

Contribution

It characterizes all matching integral traceable graphs and describes all claw-free matching integral graphs, advancing understanding of the structure of graphs with integer matching polynomial roots.

Findings

Characterization of all matching integral traceable graphs

Identification of conditions for regular matching integral graphs

Description of all claw-free matching integral graphs

Abstract

In this paper, we study graphs whose matching polynomial have only integer zeros. A graph is matching integral if the zeros of its matching polynomial are all integers. We characterize all matching integral traceable graphs.. We show that apart from K7 n (E(C3) [ E(C4)) there is no connected k-regular matching integral graph if k ? 2. It is also shown that if G is a graph with a perfect matching, then its matching polynomial has a zero in the interval (0, 1]. Finally, we describe all claw-free matching integral graphs.