Hosoya properties of power graphs over certain groups

TL;DR

This paper investigates the Hosoya properties, including the Hosoya index and polynomial, of power graphs over certain finite groups, providing insights into their structural invariants.

Contribution

It introduces the calculation of Hosoya properties specifically for power graphs of finite groups, a novel application in graph theory.

Findings

Hosoya index and polynomial are explicitly calculated for these power graphs.

The reciprocal Hosoya polynomial is also derived.

Results enhance understanding of graph invariants in algebraic structures.

Abstract

The power graph denoted by $\mathcal{P}(\mathcal{G})$ of a finite group $\mathcal{G}$ is a graph with vertex set $\mathcal{G}$ and there is an edge between two distinct elements $u, v \in \mathcal{G}$ if and only if $u^m = v$ or $v^m = u$ for some $m \in \mathbb{N}$. Depending on the distance, the Hosoya polynomial contains a lot of knowledge about graph invariants which can be used to determine well-known chemical descriptors. The Hosoya index of a graph $\Gamma$ is the total number of matchings in $\Gamma$. In this article, the Hosoya properties of the power graphs associated with a finite group, including the Hosoya index, Hosoya polynomial, and its reciprocal are calculated.