Spectral Bounds for Directed Graphs Via Asymmetric Matrices: Applications to Toughness

TL;DR

This paper extends spectral graph theory to directed graphs by establishing an Expander Mixing Lemma for asymmetric matrices, and applies it to derive bounds on graph toughness, generalizing classical results.

Contribution

It introduces a spectral bound on the toughness of directed graphs using eigenvalues of an associated asymmetric matrix, extending classical spectral inequalities.

Findings

Established an Expander Mixing Lemma for directed graphs.

Derived a spectral bound on directed graph toughness.

Generalized Alon's bound for k-regular graphs to directed graphs.

Abstract

We establish an Expander Mixing Lemma for directed graphs in terms of the eigenvalues of an associated asymmetric transition probability matrix, extending the classical spectral inequality to the asymmetric setting. As an application, we derive a spectral bound on the toughness of directed graphs that generalizes Alon's bound for $k$-regular graphs, showing how structural properties of directed graphs can be captured through their asymmetric spectra.