On the some parameters related to matching of graph powers

TL;DR

This paper investigates matching parameters in fractional powers of graphs, especially those relevant in chemistry, by analyzing natural and fractional powers of specific graphs and their properties.

Contribution

It introduces and studies matching parameters in fractional powers of graphs, focusing on graphs significant in chemistry, extending existing graph theory concepts.

Findings

Derived properties of matchings in fractional graph powers

Analyzed matching parameters for graphs relevant to chemistry

Extended understanding of graph powers in the context of matchings

Abstract

Let $G=(V,E)$ be a simple connected graph. A matching of $G$ is a set of disjoint edges of $G$. For every $n, m\in\mathbb{N}$, the $n$-subdivision of $G$ is a simple graph $G^{\frac{1}{n}}$ which is constructed by replacing each edge of $G$ with a path of length $n$ and the $m$th power of $G$, denoted by $G^m$, is a graph with the same vertex set as $G$ such that two vertices are adjacent in $G^m$ if and only if their distance is at most $m$ in $G$. The $m^{th}$ power of the $n$-subdivision of $G$ has been introduced as a fractional power of $G$ and is denoted by $G^{\frac{m}{n}}$. In this paper, we study some parameters related to matching of the natural and the fractional powers of some specific graphs. Also we study these parameters for power of graphs that are importance of in Chemistry.