Spectral Bounds for Directed Graphs Via Asymmetric Matrices: Applications to Toughness
Rebecca Carter
TL;DR
The paper addresses bounding the toughness of directed graphs using spectral methods by developing a non-Hermitian extension of the Expander Mixing Lemma tailored to directed graphs. It relies on the asymmetric transition matrix $P$, its eigenstructure, and the conditioning of the eigenbasis $\kappa(C)$ to bound discrepancies between expected and actual edge flows across vertex subsets. The main contributions include a directed Expander Mixing Lemma with explicit bounds depending on $\rho$, $\pi_{min}$, $\pi_{max}$, and $\kappa(C)$, and a spectral lower bound on toughness that generalizes Alon's bound to directed graphs. This work extends classical undirected results to directed settings without resorting to Hermitian surrogates, providing a framework to relate spectral properties to structural resilience in directed networks.
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.