Enumerating the number of $k$-matchings in successively amalgamated graphs

TL;DR

This paper introduces a transfer matrix method using the $k$-matching vector to efficiently count $k$-matchings in complex graphs formed by successive amalgamations, extending previous specialized techniques.

Contribution

The paper develops a general transfer matrix approach for counting $k$-matchings in graphs constructed via successive amalgamations, broadening the scope beyond specific chain types.

Findings

Method effectively computes $k$-matchings in complex graphs.

Extends known techniques from chemical graph theory.

Demonstrated with practical examples, including chemical structures.

Abstract

In this paper, the transfer matrix technique using the $k$-matching vector is developed to compute the number of $k$-matchings in an arbitrary graph which can be constructed by successive amalgamations over sets of cardinality two. This widely extends known methods from the literature developed for computing the number of $k$-matchings in benzenoid chains, octagonal chains, cyclooctatetraene chains, and arbitrary cyclic chains. Two examples demonstrating how the present method can be applied are given, one of then being an elaborated chemical example.