A graph without zero in its spectra

TL;DR

This paper investigates the spectral relationship between the discrete Laplacian on 1-forms and 0-forms in graphs, revealing conditions under which their non-zero spectra coincide and analyzing the 0-spectrum of the 1-form Laplacian.

Contribution

It provides new insights into the spectral properties of Laplacians on different forms, especially relating the spectra of 1-forms and 0-forms in graph theory.

Findings

Non-zero spectra of 1-form and 0-form Laplacians can coincide under certain conditions.

The 0-spectrum of the 1-form Laplacian is characterized in relation to graph cycles.

The study reveals spectral properties that exclude zero from the spectrum in specific cases.

Abstract

In this paper we consider the discrete Laplacian acting on 1-forms and we study its spectrum relative to the spectrum of the 0-form Laplacian. We show that the non zero spectrum can coincide for these Laplacians with the same nature. We examine the characteristics of 0-spectrum of the 1-form Laplacian compared to the cycles of graphs.