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

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.