تقرير
The first Steklov eigenvalue of planar graphs and beyond
| العنوان: | The first Steklov eigenvalue of planar graphs and beyond |
|---|---|
| المؤلفون: | Lin, Huiqiu, Zhao, Da |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Combinatorics, 47A75, 49J40, 49R05, 05C10 |
| الوصف: | The Steklov eigenvalue problem was introduced over a century ago, and its discrete form attracted interest recently. Let $D$ and $\delta \Omega$ be the maximum vertex degree and the set of vertices of degree one in a graph $\mathcal{G}$ respectively. Let $\lambda_2$ be the first (non-trivial) Steklov eigenvalue of $(\mathcal{G}, \delta \Omega)$. In this paper, using the circle packing theorem and conformal mapping, we first show that $\lambda_2 \leq 8D / |\delta \Omega|$ for planar graphs. This can be seen as a discrete analogue of Kokarev's bound [Variational aspects of Laplace eigenvalues on Riemannian surfaces, Adv. Math. (2014)], that is, $\lambda_2 < 8\pi / |\partial \Omega|$ for compact surfaces with boundary of genus $0$. Let $B$ and $L$ be the maximum block size and the diameter of a block graph $\mathcal{G}$ repsectively. Secondly, we prove that $\lambda_2 \leq B^2 (D-1)/ |\delta \Omega|$ and $\lambda_2 \leq (2L + (L-2)(B-2))/L^2$ for block graphs, which extend the results on trees by He and Hua [Upper bounds for the Steklov eigenvalues on trees, Calc. Var. Partial Differential Equations (2022)]. In the end, for trees with fixed leaf number and maximum degree, candidates that achieve the maximum first Steklov eigenvalue are given. Comment: 4 figures |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2407.08301 |
| رقم الأكسشن: | edsarx.2407.08301 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |