More on the sixth coefficient of the matching polynomial in regular graphs

TL;DR

This paper investigates the sixth coefficient of the matching polynomial in regular graphs and demonstrates that all cubic graphs of order 10 are uniquely identified by their matching polynomial.

Contribution

It computes the sixth coefficient of the matching polynomial for regular graphs and proves the uniqueness of cubic graphs of order 10 based on this polynomial.

Findings

Calculated the sixth coefficient for regular graphs

Proved cubic graphs of order 10 are matching unique

Enhanced understanding of matching polynomials in regular graphs

Abstract

A matching set $M$ in a graph $G$ is a collection of edges of $G$ such that no two edges from $M$ share a vertex. In this paper we consider some parameters related to the matching of regular graphs. We find the sixth coefficient of the matching polynomial of regular graphs. As a consequence, every cubic graph of order $10$ is matching unique.