Spectrum of signless 1-Laplacian on simplicial complexes

TL;DR

This paper introduces signless 1-Laplacians on simplicial complexes, exploring their spectra and connections to combinatorial properties, extending classical spectral theorems, and analyzing their behavior under topological operations.

Contribution

It presents the first study of signless 1-Laplacians on simplicial complexes, linking spectral properties to combinatorial and topological features, and extending key spectral theorems.

Findings

Spectral bounds comparable to Hoffman's bounds.

An inequality relating eigenvalue multiplicity, independence, and chromatic number.

Extension of Courant's nodal domain theorem to complexes.

Abstract

We introduce the signless 1-Laplacians and the dual Cheeger constants on simplicial complexes. The connection of its spectrum to the combinatorial properties like independence number, chromatic number and dual Cheeger constant is investigated. Our estimates can be comparable to Hoffman's bounds in virtue of Laplacian on simplicial complexes. An interesting inequality relating multiplicity of the largest eigenvalue, independence number and chromatic number are provided, which could be regarded as a variant version of Lovasz sandwich theorem. Also, the behavior of the operator under the topological operations of wedge and duplication of motifs is studied. The Courant nodal domain theorem in spectral theory is extended to the case of signless 1-Laplacian on complexes.