Spectral correspondences for finite graphs without dead ends

TL;DR

This paper explores the spectral properties of operators related to geodesic flow and its quantization on finite graphs, establishing correspondences and regularity results for eigenfunctions.

Contribution

It introduces a novel comparison between transfer and averaging operators on graphs, revealing spectral correspondences and regularity properties.

Findings

Spectral correspondences between classical and quantum operators on graphs.

Automatic regularity properties for eigenfunctions of transfer operators.

Insights into the spectral structure of graphs without dead ends.

Abstract

We compare the spectral properties of two kinds of linear operators characterizing the (classical) geodesic flow and its quantization on connected locally finite graphs without dead ends. The first kind are transfer operators acting on vector spaces associated with the set of non backtracking paths in the graphs. The second kind of operators are averaging operators acting on vector spaces associated with the space of vertices of the graph. The choice of vector spaces reflects regularity properties. Our main results are correspondences between classical and quantum spectral objects as well as some automatic regularity properties for eigenfunctions of transfer operators.