Graphs and Hermitian matrices: discrepancy and singular values

TL;DR

This paper introduces a new discrepancy measure for Hermitian matrices, establishes an inequality relating the second singular value to this discrepancy, and applies these findings to address questions about graph eigenvalues.

Contribution

It presents a novel discrepancy measure for Hermitian matrices and links it to singular values, providing new insights into graph eigenvalue problems.

Findings

Established an inequality between the second singular value and discrepancy.

Applied results to answer Fan Chung's questions on graph eigenvalues.

Provided a new analytical tool for studying Hermitian matrices and graphs.

Abstract

We introduce a measure of discrepancy of Hermitian matrices and establish an inequality between the second singular value of a Hermitian matrix and its discrepancy. These results are applied to answer two questions of Fan Chung about graph eigenvalues.