An eigenvalue localization theorem for stochastic matrices and its application to Randi\'c matrices

TL;DR

This paper establishes an eigenvalue localization theorem for stochastic matrices using principal submatrices and applies it to bound the eigenvalues of Randić matrices in connected graphs.

Contribution

It introduces a new eigenvalue localization theorem specifically for stochastic matrices and applies it to derive bounds for Randić matrix eigenvalues.

Findings

Eigenvalue bounds for Randić matrices of connected graphs

A new localization theorem for stochastic matrices

Application of submatrix analysis to spectral graph theory

Abstract

A square matrix is called stochastic (or row-stochastic) if it is non-negative and has each row sum equal to unity. Here, we constitute an eigenvalue localization theorem for a stochastic matrix, by using its principal submatrices. As an application, we provide a suitable bound for the eigenvalues, other than unity, of the Randi\'c matrix of a connected graph.