Spectrum of a class of matrices and its applications

TL;DR

This paper determines the spectrum of certain matrices using quotient matrices and applies these findings to various graph-related matrices, yielding new insights and raising open problems.

Contribution

It introduces a method to find spectra of matrices via quotient matrices and applies it to multiple graph matrices, providing new results and research directions.

Findings

Spectra of adjacency, Laplacian, and distance matrices derived

New spectral properties for graph matrices established

Open problems for future research proposed

Abstract

In this paper, we give the spectrum of a matrix by using the quotient matrix, then we apply this result to various matrices associated to a graph and a digraph, including adjacency matrix, (signless) Laplacian matrix, distance matrix, distance (signless) Laplacian matrix, to obtain some known and new results. Moreover, we propose some problems for further research.