Spectral bounds for vertex-weighted Laplacians of simplicial complexes

TL;DR

This paper provides a comprehensive spectral analysis of vertex-weighted Laplacians on simplicial complexes, establishing bounds and relations that unify and extend previous results in spectral graph theory and topology.

Contribution

It introduces new bounds and characterizations for the spectrum of vertex-weighted Laplacians, generalizing known theorems and linking spectral properties to simplicial complex operations.

Findings

Sharp upper bound on spectral radius based on vertex weights

Lower bound on spectral gap with characterization of equality cases

Explicit bounds for other eigenvalues related to weighted graphs

Abstract

The vertex-weighted Laplacian naturally extends the combinatorial Laplacian for simplicial complexes. Inspired by Lew's foundational techniques for vertex-weighted Laplacians, we present a comprehensive spectral analysis of this operator. First, we determine how basic operations, including joins, complements, and Alexander duals, affect its spectrum. This yields a sharp upper bound on the spectral radius in terms of vertex weights, along with a lower bound on the multiplicity at which this bound is attained. Second, we establish a sharp lower bound for the spectral gap and characterize when the equality holds. Third, explicit lower bounds for the remaining eigenvalues are derived, linking the vertex-weighted Laplacian spectrum to that of a related weighted graph. Finally, we reveal new spectral relations between a simplicial complex and its subcomplexes. These results not only generalize numerous known theorems on combinatorial Laplacians but also provide deeper spectral insights into simplicial structures, ultimately unifying and extending a broad range of earlier work in this field.