# Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs (I)

## TL;DR

This paper derives trace formulae for d-regular graphs that relate spectral density to periodic walks, with a tunable parameter influencing orbit contributions, linking graph spectra to random matrix theory.

## Contribution

It introduces a family of trace formulae for regular graphs with a parameter that weights different periodic orbits, connecting spectral properties to orbit structures.

## Key findings

- Trace formulae depend on a tunable parameter w.
- At w=1, only non back-scattering orbits contribute.
- The smooth spectral density matches the Kesten-McKay law at w=1.

## Abstract

Trace formulae for d-regular graphs are derived and used to express the spectral density in terms of the periodic walks on the graphs under consideration. The trace formulae depend on a parameter w which can be tuned continuously to assign different weights to different periodic orbit contributions. At the special value w=1, the only periodic orbits which contribute are the non back- scattering orbits, and the smooth part in the trace formula coincides with the Kesten-McKay expression. As w deviates from unity, non vanishing weights are assigned to the periodic walks with back-scatter, and the smooth part is modified in a consistent way. The trace formulae presented here are the tools to be used in the second paper in this sequence, for showing the connection between the spectral properties of d-regular graphs and the theory of random matrices.