The Adjacency Spectra of Some Families of Minimally Connected Prime Graphs

TL;DR

This paper investigates the spectral properties of minimally connected prime graphs using linear algebra, specifically calculating determinants and spectra for key graph families, advancing understanding in algebraic graph theory.

Contribution

It introduces a linear algebra approach to analyze prime graphs, providing explicit spectral data for important classes of minimally connected prime graphs.

Findings

Determined the determinants of adjacency matrices for certain prime graph families.

Calculated the spectra of key minimally connected prime graphs.

Enhanced the algebraic understanding of prime graphs in group theory.

Abstract

In finite group theory, studying the prime graph of a group has been an important topic for almost the past half-century. Recently, prime graphs of solvable groups have been characterized in graph theoretical terms only. This now allows the study of these graphs without any knowledge of the group theoretical background. In this paper we study prime graphs from a linear algebra angle and focus on the class of minimally connected prime graphs introduced in earlier work on the subject. As our main results, we determine the determinants of the adjacency matrices and the spectra of some important families of these graphs.