Finite projective planes meet spectral gaps
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909816369184768 |
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| author | Guo, Yuhan Zhang, Dong |
| author_facet | Guo, Yuhan Zhang, Dong |
| contents | We show that for any connected graph $G$ with maximum degree $d\ge3$, the spectral gap from $0$ with respect to the adjacency matrix is at most $\sqrt{d-1}$, with equality if and only if $G$ is the incidence graph of a finite projective plane of order $d-1$; and for other cases, the bound $\sqrt{d-1}$ is improved to $\sqrt{d-2}$. This is a spectral gap version of a result by Mohar and Tayfeh-Rezaie. Moreover, for $d$-regular graphs with girth at least 7, the bound $\sqrt{d-2}$ is further improved to $\sqrt{d-c(d)}$ where $c(d)\ge 2$ and $\lim\limits_{d\to\infty}c(d)/d=(\sqrt{5}-1)/2$.
A similar yet more subtle phenomenon involving the normalized Laplacian is also investigated, where we work on graphs of degrees $\ge d$ rather than $\le d$. We prove that for any graph $G$ with \emph{minimum} degree $d\ge 3$, the spectral gap from the value 1 with respect to the normalized Laplacian is at most $\sqrt{d-1}/d$, with equality if and only if $G$ is the incidence graph of a finite projective plane of order $d-1$. As an application, we provide a new sharp bound for the convergence rate of some eigenvalues of the Laplacian on the weighted neighborhood graphs introduced by Bauer and Jost. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06247 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finite projective planes meet spectral gaps Guo, Yuhan Zhang, Dong Combinatorics Spectral Theory We show that for any connected graph $G$ with maximum degree $d\ge3$, the spectral gap from $0$ with respect to the adjacency matrix is at most $\sqrt{d-1}$, with equality if and only if $G$ is the incidence graph of a finite projective plane of order $d-1$; and for other cases, the bound $\sqrt{d-1}$ is improved to $\sqrt{d-2}$. This is a spectral gap version of a result by Mohar and Tayfeh-Rezaie. Moreover, for $d$-regular graphs with girth at least 7, the bound $\sqrt{d-2}$ is further improved to $\sqrt{d-c(d)}$ where $c(d)\ge 2$ and $\lim\limits_{d\to\infty}c(d)/d=(\sqrt{5}-1)/2$. A similar yet more subtle phenomenon involving the normalized Laplacian is also investigated, where we work on graphs of degrees $\ge d$ rather than $\le d$. We prove that for any graph $G$ with \emph{minimum} degree $d\ge 3$, the spectral gap from the value 1 with respect to the normalized Laplacian is at most $\sqrt{d-1}/d$, with equality if and only if $G$ is the incidence graph of a finite projective plane of order $d-1$. As an application, we provide a new sharp bound for the convergence rate of some eigenvalues of the Laplacian on the weighted neighborhood graphs introduced by Bauer and Jost. |
| title | Finite projective planes meet spectral gaps |
| topic | Combinatorics Spectral Theory |
| url | https://arxiv.org/abs/2509.06247 |