The largest Laplacian eigenvalue and the balancedness of simplicial complexes

TL;DR

This paper investigates the spectral properties of Laplacians in simplicial complexes, establishing bounds on eigenvalues and characterizing balancedness through combinatorial and topological operations.

Contribution

It introduces new bounds for the largest eigenvalue of up Laplacians, characterizes balancedness via eigenvalues, and explores how complex operations affect these properties.

Findings

Largest eigenvalue of up Laplacian is bounded by that of signless Laplacian.

Equality in eigenvalues characterizes balancedness of the incidence graph.

Constructs infinitely many complexes with balanced incidence graphs using wedge sum and motifs.

Abstract

Let $K$ be a simplical complex, and let $\mathcal{L}_i^{up}(K), \mathcal{Q}_i^{up}(K)$ be the $i$-th up Laplacian and signless Laplacian of $K$, respectively. In this paper we proved that the largest eigenvalue of $\mathcal{L}_i^{up}(K)$ is not greater than the largest eigenvalue of $\mathcal{Q}_i^{up}(K)$; furthermore, if $K$ is $(i+1)$-path connected, then the equality holds if and only if the $i$-th incidence signed graph $B_i(K)$ of $K$ is balanced. As an application we provided an upper bound for the largest eigenvalue of the $i$-th up Laplacian of $K$, which improves the bound given by Horak and Jost and generalizes the result of Anderson and Morley on graphs.We characterized the balancedness of simplicial complexes under operations such as wedge sum, join, Cartesian product and duplication of motifs. For each $i \ge 0$, by using wedge sum or duplication of motifs, we can construct an infinitely many $(i+1)$-path connected simplicial complexes $K$ with $B_i(K)$ being balanced.