Equivalence between spectral properties of graphs with and without loops

TL;DR

This paper establishes a spectral equivalence between graphs with loops and loopless graphs by generalizing spectral results and defining a covering space, enabling analysis of graphs with loops as simpler, loopless graphs.

Contribution

It introduces a spectra-preserving relation between graphs with and without loops, extending spectral results to a broader class and defining a covering space to remove loops while preserving spectral properties.

Findings

Spectral properties of graphs with loops can be studied via equivalent loopless graphs.

Generalization of spectral results from (m, k)-stars to (m, k, s)-stars.

A covering space construction preserves spectra while removing loops.

Abstract

In this paper we introduce a spectra preserving relation between graphs with loops and graphs without loops. This relation is achieved in two steps. First, by generalizing spectra results got on (m, k)-stars to a wider class of graphs, the (m, k, s)-stars with or without loops. Second, by defining a covering space of graphs with loops that allows to remove the presence of loops by increasing the graph dimension. The equivalence of the two class of graphs allows to study graph with loops as simple graph without loosing information.