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

Abstract:

A non-backtracking walk on a graph, $H$, is a directed path of directed edges of $H$ 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, $B$, indexed by $H$’s directed edges and related to Ihara’s Zeta function.

We show how to determine $B$’s spectrum in the case where $H$ is a tree covering a finite graph. We show that when $H$ is not regular, this spectrum can have positive measure in the complex plane, unlike the regular case. We show that outside of $B$’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 $\sqrt {\mathrm {gr}}$, where $\mathrm {gr}$ is the cogrowth of $B$, 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, $H$, a random lift of large degree has non-backtracking new spectrum near that of $H$’s universal cover. This suggests a new generalization of Alon’s second eigenvalue conjecture.