The number of maximal matchings in polygon rings

TL;DR

This paper derives a formula for counting maximal matchings in polygon rings using matrix traces and extends the method to arbitrary polygon rings with an algorithm for transition matrices.

Contribution

It introduces a novel matrix-based approach to count maximal matchings in polygon rings and provides an algorithm for transition matrices in polygon chains.

Findings

Number of maximal matchings equals the trace of a product of specific matrices.

Method applies to hexagonal and arbitrary polygon rings.

An algorithm for determining transition matrices is provided.

Abstract

A matching of graph $G$ is maximal if it cannot be expanded by adding any edge to create a larger matching. In this paper, for a hexagonal ring $H$ with $n$ hexagons, we show that the number of maximal matchings of $H$ equals to the trace of the product of $n$ matrices, each of which is $S$, $L$, or $R$ according to the type of the connection mode of $H$. Finally, we extend this conclusion to arbitrary polygon rings and provide an algorithm to determine the transition matrices of polygon chains (rings).