Laplacian eigenvalues of independence complexes via additive compound matrices
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912160452444160 |
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| author | Lew, Alan |
| author_facet | Lew, Alan |
| contents | 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 \[ \left| \left\{ 1\leq i_1<\cdots<i_{k+1}\leq |V| : \, λ_{i_1}+λ_{i_2}+\cdots+λ_{i_{k+1}} \geq |V|\right\}\right|,\] where $λ_1\geqλ_2\geq \cdots\geq λ_{|V|}=0$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_14496 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Laplacian eigenvalues of independence complexes via additive compound matrices Lew, Alan Combinatorics 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 \[ \left| \left\{ 1\leq i_1<\cdots<i_{k+1}\leq |V| : \, λ_{i_1}+λ_{i_2}+\cdots+λ_{i_{k+1}} \geq |V|\right\}\right|,\] where $λ_1\geqλ_2\geq \cdots\geq λ_{|V|}=0$ 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. |
| title | Laplacian eigenvalues of independence complexes via additive compound matrices |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2307.14496 |