The Non-Backtracking Spectrum of the Universal Cover of a Graph

Understand

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.