Hosoya properties of power graphs over certain groups

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.