The first Steklov eigenvalue of planar graphs and beyond
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915362904211456 |
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| author | Lin, Huiqiu Zhao, Da |
| author_facet | Lin, Huiqiu Zhao, Da |
| contents | The Steklov eigenvalue problem was introduced over a century ago, and its discrete form attracted interest recently. Let $D$ and $δΩ$ be the maximum vertex degree and the set of vertices of degree one in a graph $\mathcal{G}$ respectively. Let $λ_2$ be the first (non-trivial) Steklov eigenvalue of $(\mathcal{G}, δΩ)$. In this paper, using the circle packing theorem and conformal mapping, we first show that $λ_2 \leq 8D / |δΩ|$ for planar graphs. This can be seen as a discrete analogue of Kokarev's bound, that is, $λ_2 < 8π/ |\partial Ω|$ for compact surfaces with boundary of genus $0$. Let $B$ and $L$ be the maximum block size and the diameter of a block graph $\mathcal{G}$ respectively. Secondly, we prove that $λ_2 \leq 4 (B-1) (D-1)/ |δΩ|$ and $λ_2 \leq B/L$ for block graphs, which extend the results on trees by He and Hua. In the end, for trees with fixed leaf number and maximum degree, candidates that achieve the maximum first Steklov eigenvalue are given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08301 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The first Steklov eigenvalue of planar graphs and beyond Lin, Huiqiu Zhao, Da Combinatorics 47A75, 49J40, 49R05, 05C10 The Steklov eigenvalue problem was introduced over a century ago, and its discrete form attracted interest recently. Let $D$ and $δΩ$ be the maximum vertex degree and the set of vertices of degree one in a graph $\mathcal{G}$ respectively. Let $λ_2$ be the first (non-trivial) Steklov eigenvalue of $(\mathcal{G}, δΩ)$. In this paper, using the circle packing theorem and conformal mapping, we first show that $λ_2 \leq 8D / |δΩ|$ for planar graphs. This can be seen as a discrete analogue of Kokarev's bound, that is, $λ_2 < 8π/ |\partial Ω|$ for compact surfaces with boundary of genus $0$. Let $B$ and $L$ be the maximum block size and the diameter of a block graph $\mathcal{G}$ respectively. Secondly, we prove that $λ_2 \leq 4 (B-1) (D-1)/ |δΩ|$ and $λ_2 \leq B/L$ for block graphs, which extend the results on trees by He and Hua. In the end, for trees with fixed leaf number and maximum degree, candidates that achieve the maximum first Steklov eigenvalue are given. |
| title | The first Steklov eigenvalue of planar graphs and beyond |
| topic | Combinatorics 47A75, 49J40, 49R05, 05C10 |
| url | https://arxiv.org/abs/2407.08301 |