Edge Laplacians and Edge Poisson Transforms for Graphs

TL;DR

This paper establishes a quantum-classical correspondence for edge Laplacians on graphs, relating spectral properties to topological features via edge Poisson transforms and transfer operators, extending previous vertex-based results.

Contribution

It introduces a novel quantum-classical correspondence for edge Laplacians valid for all spectral parameters, linking spectral data to graph topology through edge Poisson transforms.

Findings

Isomorphisms between edge Laplacian eigenspaces and transfer operator eigenspaces

Relation of spectral quantities to graph topological properties

Connection of edge Poisson transforms to automorphism group representations in regular trees

Abstract

For a finite graph, we establish natural isomorphisms between eigenspaces of a Laplace operator acting on functions on the edges and eigenspaces of a transfer operator acting on functions on one-sided infinite non-backtracking paths. Interpreting the transfer operator as a classical dynamical system and the Laplace operator as its quantization, this result can be viewed as a quantum-classical correspondence. In contrast to previously established quantum-classical correspondences for the vertex Laplacian which exclude certain exceptional spectral parameters, our correspondence is valid for all parameters. This allows us to relate certain spectral quantities to topological properties of the graph such as the cyclomatic number and the 2-colorability. The quantum-classical correspondence for the edge Laplacian is induced by an edge Poisson transform on the universal covering of the graph which is a tree of bounded degree. In the special case of regular trees, we relate both the vertex and the edge Poisson transform to the representation theory of the automorphism group of the tree and study associated operator valued Hecke algebras.