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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.26251 |
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| _version_ | 1866914426552057856 |
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| author | Ai, Jiangdong Ji, Yizhe Lian, Xiaopan Yang, Kun |
| author_facet | Ai, Jiangdong Ji, Yizhe Lian, Xiaopan Yang, Kun |
| contents | Let $T$ be a finite tree with leaf set $\dO$ as the boundary and let $λ_2$ be the first nontrivial Steklov eigenvalue. Let $D$ and $\ell$ be the maximum vertex degree and the number of leaves, respectively. Motivated by the spectral influence of neck-stretching on Riemannian manifolds, we investigate a discrete counterpart--edge-stretching--and its effect on the Steklov eigenvalues of graphs. We prove that Steklov eigenvalues decrease monotonically under the edge--stretching operation. As a consequence, we prove that $λ_2\le D/\ell$, with equality if and only if $T$ is a star. This fundamentally improves the constant in He--Hua's bound $λ_2\le 4(D-1)/\ell$ to the optimal value~$1$.
We also provide a closed-form diagonalization of the Steklov problem on level--regular trees, yielding explicit eigenvalues and multiplicities. In addition, we provide a general upper bound $λ_k\le \min\{1,\,16Dk/\ell\}$ for higher eigenvalues. Systematic numerical experiments verify the sharp bound and provide evidence for the extremal conjecture of Lin--Zhao on balanced minimum--height trees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_26251 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Effect of edge-stretching on Steklov eigenvalues and sharp Steklov eigenvalue bounds on leaf--boundary trees Ai, Jiangdong Ji, Yizhe Lian, Xiaopan Yang, Kun Combinatorics Let $T$ be a finite tree with leaf set $\dO$ as the boundary and let $λ_2$ be the first nontrivial Steklov eigenvalue. Let $D$ and $\ell$ be the maximum vertex degree and the number of leaves, respectively. Motivated by the spectral influence of neck-stretching on Riemannian manifolds, we investigate a discrete counterpart--edge-stretching--and its effect on the Steklov eigenvalues of graphs. We prove that Steklov eigenvalues decrease monotonically under the edge--stretching operation. As a consequence, we prove that $λ_2\le D/\ell$, with equality if and only if $T$ is a star. This fundamentally improves the constant in He--Hua's bound $λ_2\le 4(D-1)/\ell$ to the optimal value~$1$. We also provide a closed-form diagonalization of the Steklov problem on level--regular trees, yielding explicit eigenvalues and multiplicities. In addition, we provide a general upper bound $λ_k\le \min\{1,\,16Dk/\ell\}$ for higher eigenvalues. Systematic numerical experiments verify the sharp bound and provide evidence for the extremal conjecture of Lin--Zhao on balanced minimum--height trees. |
| title | Effect of edge-stretching on Steklov eigenvalues and sharp Steklov eigenvalue bounds on leaf--boundary trees |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.26251 |