Discrete trace formulas and holomorphic functional calculus for the adjacency matrix of regular graphs

TL;DR

This paper introduces a unified approach using holomorphic functional calculus to analyze the spectra of regular graphs' adjacency matrices, deriving trace formulas and providing new proofs for classical problems in spectral graph theory.

Contribution

It develops a systematic method to connect spectral properties with combinatorial graph features via non-backtracking matrices and holomorphic calculus.

Findings

Derived discrete trace formulas linking spectrum and graph structure

Provided new proofs for walk counting and Ihara-Bass formula

Unified framework for heat and Schrödinger equations on graphs

Abstract

We provide a unified method to study the adjacency matrices of regular graphs (including infinite ones) using holomorphic functional calculus. By applying this calculus on a specific ellipse that contains the spectrum, we derive an expansion of $h(A)$ using non-backtracking matrices. This framework allows us to systematically obtain discrete trace formulas that link spectral theory with graph combinatorics. To show how this method works, we give new proofs for several well-known problems, such as walk counting, the Ihara-Bass formula, and solutions to the heat and Schr\"odinger equations on graphs.