Spectra of sparse regular graphs with loops

TL;DR

This paper derives exact spectral equations for sparse regular graphs with loops, revealing how loop length influences spectral properties and network dynamics such as synchronization.

Contribution

It provides novel analytical formulas for the spectra of regular graphs with loops of arbitrary length, enhancing understanding of their structural and dynamical features.

Findings

Loops affect the spectral gap size

Spectral formulas depend on loop length

Results inform network synchronization properties

Abstract

We derive exact equations that determine the spectra of undirected and directed sparsely connected regular graphs containing loops of arbitrary length. The implications of our results to the structural and dynamical properties of networks are discussed by showing how loops influence the size of the spectral gap and the propensity for synchronization. Analytical formulas for the spectrum are obtained for specific length of the loops.