On the spectra of large sparse graphs with cycles

TL;DR

This paper introduces a statistical mechanics-based method to accurately compute the spectra of large sparse graphs with cycles, providing exact formulas for regular graphs and their spectral boundaries.

Contribution

It develops a novel approach for spectral analysis of graphs with cycles, deriving exact equations and analytical formulas for their spectra.

Findings

Derived exact equations for the resolvent of adjacency matrices.

Obtained analytical formulas for spectra and spectral boundaries.

Achieved high-accuracy spectral computations for large graphs.

Abstract

We present a general method for obtaining the spectra of large graphs with short cycles using ideas from statistical mechanics of disordered systems. This approach leads to an algorithm that determines the spectra of graphs up to a high accuracy. In particular, for (un)directed regular graphs with cycles of arbitrary length we derive exact and simple equations for the resolvent of the associated adjacency matrix. Solving these equations we obtain analytical formulas for the spectra and the boundaries of their support.