Upper bounds of Steklov eigenvalues on graphs
| Autor: | Lin, Huiqiu, Liu, Lianping, You, Zhe, Zhao, Da |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Let $\Delta$ and $B$ be the maximum vertex degree and a subset of vertices in a graph $G$ respectively. In this paper, we study the first (non-trivial) Steklov eigenvalue $\sigma_2$ of $G$ with boundary $B$. Using metrical deformation via flows, we first show that $\sigma_2 = \mathcal{O}\left(\frac{\Delta(g+1)^3}{|B|}\right)$ for graphs of orientable genus $g$ if $|B| \geq \max\{3 \sqrt{g},|V|^{\frac{1}{4} + \epsilon}, 9\}$ for some $\epsilon > 0$. This can be seen as a discrete analogue of Karpukhin's bound. Secondly, we prove that $\sigma_2 \leq \frac{8\Delta+4X}{|B|}$ based on planar crossing number $X$. Thirdly, we show that $\sigma_2 \leq \frac{|B|}{|B|-1} \cdot \delta_B$, where $\delta_B$ denotes the minimum degree for boundary vertices in $B$. At last, we compare several upper bounds on Laplacian eigenvalues and Steklov eigenvalues. Comment: 22 pages, 2 figures |
| Databáze: | arXiv |
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