The second largest eigenvalue of a transition matrix @math has connections with many properties of the underlying Markov chain, and especially its convergence rate towards the stationary distribution. In this paper, we give an asymptotic upper bound for the second eigenvalue when @math is the transition matrix of the simple random walk over a random directed graph with given degree sequence. This is the first result concerning the asymptotic behavior of the spectral gap for sparse non-reversible Markov chains with an unknown stationary distribution. An immediate consequence of our result is a generalization of the well-known Friedman theorem, for undirected regular graphs. Our result is based on a variation of the trace method introduced by Bordenave (2015).
Abstract It is shown that the second largest eigenvalue of the adjacency matrix of any d-regular graph G containing two edges the distance between which is at least 2k + 2 is at least 2 d − 1 − (2 d − 1 − 1) (k+1) .