# Spectral Analysis for Non-Hermitian Matrices and Directed Graphs

## TL;DR

This paper extends spectral graph theory to non-Hermitian matrices and directed graphs, introducing new conditions and bounds that broaden the applicability of spectral methods.

## Contribution

It introduces admissibility conditions and variational estimates for non-Hermitian matrices, enabling spectral analysis of directed graphs.

## Key findings

- New bounds on independent set size in directed graphs
- Generalization of spectral estimates to non-Hermitian matrices
- Applicable techniques for analyzing non-Hermitian systems

## Abstract

We generalize classical results in spectral graph theory and linear algebra more broadly, from the case where the underlying matrix is Hermitian to the case where it is non-Hermitian. New admissibility conditions are introduced to replace the Hermiticity condition. We prove new variational estimates of the Rayleigh quotient for non-Hermitian matrices. As an application, a new Delsarte-Hoffman-type bound on the size of the largest independent set in a directed graph is developed. Our techniques consist in quantifying the impact of breaking the Hermitian symmetry of a matrix and are broadly applicable.