Laplacian eigenvalues of independence complexes via additive compound matrices

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.