Proof of Lew's conjecture on the spectral gaps of simplicial complexes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916338547556352 |
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| author | Zhan, Xiongfeng Huang, Xueyi Lin, Huiqiu |
| author_facet | Zhan, Xiongfeng Huang, Xueyi Lin, Huiqiu |
| contents | As a generalization of graph Laplacians to higher dimensions, the combinatorial Laplacians of simplicial complexes have garnered increasing attention. Let $X$ be a simplicial complex on vertex set $V$ of size $n$, and let $X(k)$ denote the set of all $k$-dimensional simplices of $X$. The $k$-th spectral gap $μ_k(X)$ is the smallest eigenvalue of the reduced $k$-dimensional Laplacian of $X$. For any $k\geq -1$, Lew [J. Combin. Theory Ser. A 169 (2020) 105127] established a lower bound for $μ_k(X)$: $$μ_k(X)\geq (d+1)\left(\min_{σ\in X(k)}°_X(σ)+k+1\right)-dn\geq (d+1)(k+1)-dn,$$ where $°_X(σ)$ and $d$ denote the degree of $σ$ in $X$ and the maximal dimension of a missing face of $X$, respectively. In this paper, we identify the unique simplicial complex that achieves the lower bound of the $k$-th spectral gap, $(d+1)(k+1)-dn$, for some $k$, thereby confirming a conjecture proposed by Lew. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_10398 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Proof of Lew's conjecture on the spectral gaps of simplicial complexes Zhan, Xiongfeng Huang, Xueyi Lin, Huiqiu Combinatorics 05E45 As a generalization of graph Laplacians to higher dimensions, the combinatorial Laplacians of simplicial complexes have garnered increasing attention. Let $X$ be a simplicial complex on vertex set $V$ of size $n$, and let $X(k)$ denote the set of all $k$-dimensional simplices of $X$. The $k$-th spectral gap $μ_k(X)$ is the smallest eigenvalue of the reduced $k$-dimensional Laplacian of $X$. For any $k\geq -1$, Lew [J. Combin. Theory Ser. A 169 (2020) 105127] established a lower bound for $μ_k(X)$: $$μ_k(X)\geq (d+1)\left(\min_{σ\in X(k)}°_X(σ)+k+1\right)-dn\geq (d+1)(k+1)-dn,$$ where $°_X(σ)$ and $d$ denote the degree of $σ$ in $X$ and the maximal dimension of a missing face of $X$, respectively. In this paper, we identify the unique simplicial complex that achieves the lower bound of the $k$-th spectral gap, $(d+1)(k+1)-dn$, for some $k$, thereby confirming a conjecture proposed by Lew. |
| title | Proof of Lew's conjecture on the spectral gaps of simplicial complexes |
| topic | Combinatorics 05E45 |
| url | https://arxiv.org/abs/2407.10398 |