Joins of normal matrices, their spectrum, and applications

TL;DR

This paper investigates the spectral properties of the join of normal matrices with constant row sums, applying findings to spectral graph theory, including constructions of Ramanujan graphs and new proofs of classical theorems.

Contribution

It introduces new methods for analyzing the spectrum of matrix joins and applies these to derive results in spectral graph theory and graph constructions.

Findings

Derived the characteristic polynomial for joins of regular graphs

Provided new constructions of Ramanujan graphs

Presented alternative proofs for classical spectral theorems

Abstract

Motivated by studies of oscillator networks, we study the spectrum of the join of several normal matrices with constant row sums. We apply our results to compute the characteristic polynomial of the join of several regular graphs. We then use this theorem to study several problems in spectral graph theory. In particular, we provide some simple constructions of Ramanujan graphs and give new proofs for some theorems in the classical book of Cvetkovi\'{c}, Rowlinson, and Slobodan.