The Szeged Index of Power Graph of Finite Groups

TL;DR

This paper derives formulas for the Szeged index of power graphs of finite groups, especially cyclic and dihedral groups, and provides computational tools for these indices.

Contribution

It introduces a formula for the Szeged index of the generalized join of graphs and applies it to power graphs of finite cyclic and dihedral groups.

Findings

Szeged index formula for the generalized join of graphs

Explicit Szeged index for power graph of cyclic groups

Relation between Szeged indices of cyclic and dihedral groups

Abstract

The Szeged index of a graph is an invariant with several applications in chemistry. The power graph of a finite group $G$ is a graph having vertex set as $G$ in which two vertices $u$ and $v$ are adjacent if $v=u^m$ or $u=v^n$ for some $m,n\in \mathbb{N}$. In this paper, we first obtain a formula for the Szeged index of the generalized join of graphs. As an application, we obtain the Szeged index of the power graph of the finite cyclic group $\mathbb Z_n$ for any $n>2$. We further obtain a relation between the Szeged index of the power graph of $\mathbb Z_n$ and the Szeged index of the power graph of the dihedral group $\mathrm{D}_n$. We also provide SAGE codes for evaluating the Szeged index of the power graph of $\mathbb{Z}_n$ and $\mathrm{D}_n$ at the end of this paper.