Prime Order Element Graph of a Group -- II

TL;DR

This paper extends the study of the prime order element graph of finite groups by analyzing its spectral properties, planarity, and clique number, providing new insights and outlining future research directions.

Contribution

It offers a detailed spectral and structural analysis of the prime order element graph, advancing understanding beyond previous work.

Findings

Spectral properties of the adjacency and Laplacian matrices analyzed.

Conditions for planarity and clique number determined.

Open problems and future research directions outlined.

Abstract

In this sequel paper, we continue the analysis of the prime order element graph $\Gamma(G)$ of a finite group $G$, where vertices are elements of $G$ and edges connect distinct elements $x, y$ satisfying $\circ(xy) = p$ for some prime $p$. Our investigation focuses on the adjacency and Laplacian spectra, planarity, and clique number of this graph. We conclude by outlining open issues and potential directions for future investigations.