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The adjacency spectra of some families of minimally connected prime graphs

Chris Florez, Jonathan Higgins, Kyle Huang, Thomas Michael Keller and Dawei Shen

Vol. 17 (2024), No. 1, 107–120
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. We approach prime graphs from a linear-algebraic angle and focus on the class of minimally connected prime graphs introduced in earlier work on the subject. As our main results, we prove new properties about the adjacency matrices of some special families of these graphs, focusing on their characteristic polynomials and spectra.

Keywords
prime graphs, adjacency matrix, spectral graph theory
Mathematical Subject Classification
Primary: 05C25, 15A18
Secondary: 20D10
Milestones
Received: 29 August 2022
Revised: 10 December 2022
Accepted: 17 December 2022
Published: 15 March 2024

Communicated by Vadim Ponomarenko
Authors
Chris Florez
Department of Mathematics
David Rittenhouse Lab
University of Pennsylvania
Philadelphia, PA
United States
Jonathan Higgins
Department of Mathematics
University of Illinois Urbana-Champaign
Urbana, IL
United States
Kyle Huang
Department of Mathematics
Freie Universität Berlin
Berlin
Germany
Thomas Michael Keller
Department of Mathematics
Texas State University
San Marcos, TX
United States
Dawei Shen
Department of Mathematics
University of Michigan
Ann Arbor, MI
United States