The non-backtracking spectrum of the universal cover of a graph

Abstract

A non-backtracking walk on a graph, H H , is a directed path of directed edges of H H such that no edge is the inverse of its preceding edge. Non-backtracking walks of a given length can be counted using the non-backtracking adjacency matrix, B B , indexed by H H ’s directed edges and related to Ihara’s Zeta function. We show how to determine B B ’s spectrum in the case where H H is a tree covering a finite graph. We show that when H H is not regular, this spectrum can have positive measure in the complex plane, unlike the regular case. We show that outside of B B ’s spectrum, the corresponding Green function has “periodic decay ratios”. The existence of such a “ratio system” can be effectively checked and is equivalent to being outside the spectrum. We also prove that the spectral radius of the non-backtracking walk operator on the tree covering a finite graph is exactly g r \sqrt {\mathrm {gr}} , where g r \mathrm {gr} is the cogrowth of B B , or growth rate of the tree. This further motivates the definition of the graph theoretical Riemann hypothesis proposed by Stark and Terras. Finally, we give experimental evidence that for a fixed, finite graph, H H , a random lift of large degree has non-backtracking new spectrum near that of H H ’s universal cover. This suggests a new generalization of Alon’s second eigenvalue conjecture.