Laplacian eigenvalues of independence complexes via additive compound matrices

TL;DR

This paper establishes new bounds on the eigenvalues of Laplacians of independence complexes of graphs, linking topological properties to spectral graph theory through additive compound matrices.

Contribution

It introduces a novel relation between Laplacian eigenvalues of independence complexes and additive compound matrices, extending previous homology bounds.

Findings

Derived lower bounds on Laplacian eigenvalues of independence complexes.

Connected eigenvalue sums to vanishing of homology groups.

Extended results to vertex-weighted Laplacians.

Abstract

The independence complex of a graph $G=(V,E)$ is the simplicial complex $I(G)$ on vertex set $V$ whose simplices are the independent sets in $G$. We present new lower bounds on the eigenvalues of the $k$-dimensional Laplacian $L_k(I(G))$ in terms of the eigenvalues of the graph Laplacian $L(G)$. As a consequence, we show that for all $k\geq 0$, the dimension of the $k$-th reduced homology group (with real coefficients) of $I(G)$ is at most \[ \left| \left\{ 1\leq i_1<\cdots<i_{k+1}\leq |V| : \, \lambda_{i_1}+\lambda_{i_2}+\cdots+\lambda_{i_{k+1}} \geq |V|\right\}\right|,\] where $\lambda_1\geq\lambda_2\geq \cdots\geq \lambda_{|V|}=0$ are the eigenvalues of $L(G)$. In particular, if $k$ is the minimal number such that the sum of the $k$ largest eigenvalues of $L(G)$ is at least $|V|$, then $\tilde{H}_i(I(G);\mathbb{R})=0$ for all $i\leq k-2$. This extends previous results by Aharoni, Berger and Meshulam. Our proof relies on a relation between the $k$-dimensional Laplacian $L_k(I(G))$ and the $(k+1)$-th additive compound matrix of $L_0(I(G))$, which is an $\binom{n}{k+1}\times\binom{n}{k+1}$ matrix whose eigenvalues are all the possible sums of $k+1$ eigenvalues of the $0$-dimensional Laplacian. Our results apply also in the more general setting of vertex-weighted Laplacian matrices.