Eigenvalue bounds for the signless $p$-Laplacian

TL;DR

This paper investigates the spectral properties of the signless p-Laplacian of graphs, establishing bounds for eigenvalues and revealing behaviors as p approaches 1, extending classical results.

Contribution

It introduces new bounds for the eigenvalues of the signless p-Laplacian, generalizing previous bipartiteness bounds and analyzing the limit as p approaches 1.

Findings

Largest eigenvalue has Perron-Frobenius property

Derived upper and lower bounds for the smallest eigenvalue

Bounds converge as p approaches 1

Abstract

We consider the signless $p$-Laplacian of a graph, a generalisation of the usual signless Laplacian (the case $p=2$). We show a Perron-Frobenius property and basic inequalites for the largest eigenvalue and provide upper and lower bounds for the smallest eigenvalue in terms of a parameter related to the bipartiteness. The latter result generalises bounds by Desai and Rao and, interestingly, in the limit $p\to 1$ upper and lower bounds coincide.