Hosoya Polynomials of Power Graphs of Certain Finite Groups

Abstract

Assume that G is a finite group. The power graph P(G) of G is a graph in which G is its node set, where two different elements are connected by an edge whenever one of them is a power of the other. A topological index is a number generated from a molecular structure that indicates important structural properties of the proposed molecule. Indeed, it is a numerical quantity connected with the chemical composition that is used to correlate chemical structures with various physical characteristics, chemical reactivity, and biological activity. This information is important for identifying well-known chemical descriptors based on distance dependence. In this paper, we study Hosoya properties, such as the Hosoya polynomial and the reciprocal status Hosoya polynomial of power graphs of various finite cyclic and non-cyclic groups of order pq and pqr, where p,q and r(p≥q≥r) are prime numbers.

Keywords: Hosoya polynomial; finite groups; molecular structure; power graphs.

Figure 1

$\mathcal{P}({\mathbb{Z}}_r\times F_{p,q}),(p\cong 1\left(\mathrm{mod}q\right))$.