Maximal matchings in polyspiro and benzenoid chains

TL;DR

This paper develops recurrence relations and generating functions to enumerate maximal matchings in polyspiro and benzenoid chains, analyzing their asymptotic behavior and extremal cases, thus advancing understanding of these less-studied graph matchings.

Contribution

It introduces the first known recurrences and generating functions for counting maximal matchings in specific chemical graph classes, with asymptotic analysis included.

Findings

Derived recurrences for maximal matchings in polyspiro and benzenoid chains

Established generating functions for these sequences

Analyzed asymptotic behavior and identified extremal cases

Abstract

A matching $M$ of a graph $G$ is maximal if it is not a proper subset of any other matching in $G$. Maximal matchings are much less known and researched than their maximum and perfect counterparts. In particular, almost nothing is known about their enumerative properties. In this paper we present the recurrences and generating functions for the sequences enumerating maximal matchings in two classes of chemically interesting linear polymers: polyspiro chains and benzenoid chains. We also analyze the asymptotic behavior of those sequences and determine the extremal cases.