On the Number of $k-$Matchings in Graphs

TL;DR

This paper derives exact formulas for counting 3-, 4-, and 5-matchings in simple graphs using degree-based invariants, advancing combinatorial graph enumeration methods.

Contribution

It provides new exact formulas for the number of k-matchings in graphs for k=3,4,5 based on degree invariants, filling a gap in combinatorial graph theory.

Findings

Formulas for p(G,3), p(G,4), p(G,5) derived

Expressed in terms of degree-based invariants

Enhances understanding of graph matchings

Abstract

Suppose $G$ is a undirected simple graph. A $k-$subset of edges in $G$ without common vertices is called a $k-$matching and the number of such subsets is denoted by $p(G,k)$. The aim of this paper is to present exact formulas for $p(G,3)$, $p(G,4)$ and $P(G,5)$ in terms of some degree-based invariants.