AI 中文总结
本文建立了Burger-Brooks转移原理的新变体，得到图的Steklov特征值的上界，解决了Lin和Zhao提出的问题，还改进了图的拉普拉斯特征值的已有上界。
AI 中文摘要
本文建立了Burger-Brooks转移原理的新变体，该原理允许我们应用带测度的黎曼曲面的谱估计，得到如下结果：存在一个通用常数$C>0$，使得对于每一个带边界$B$、最大度$d_{\max}$和亏格$g$的连通图$G=(V,E)$，当$1\leq k\leq |B|$时，有$\sigma_k(G,B)\leq C d_{\max}\frac{g+k}{|B|}$，其中$\sigma_k(G,B)$表示带边界$B$的图$G$的第$k$个Steklov特征值。该界在通用常数范围内是最优的，从而解决了Lin和Zhao[J. Lond. Math. Soc. (2) 112 (2025), Paper No. e70238]提出的问题。此外，当$B=V$时，上述结果给出了图的拉普拉斯特征值的上界，改进了Kelner、Lee、Price和Teng[Geom. Funct. Anal. 21 (2011), 1117--1143]以及Amini和Cohen-Steiner[Comment. Math. Helv. 93 (2018), 203--223]之前已知的界。
英文摘要
In this paper, we establish a new variant of the Burger-Brooks transfer principle, which allows us to apply spectral estimates for measured Riemannian surfaces to obtain the following result: There exists a universal constant $C>0$ such that, for every connected graph $G=(V, E)$ with boundary $B$, maximum degree $d_{\max}$ and genus $g$, \[σ_k(G, B)\leq C d_{\max}\frac{g+k}{|B|},\] where $1\leq k\leq |B|$ and $σ_k(G, B)$ denotes the $k$-th Steklov eigenvalue of $G$ with boundary $B$. This bound is sharp up to a universal constant, thereby resolving a problem raised by Lin and Zhao [J. Lond. Math. Soc. (2) 112 (2025), Paper No. e70238]. Furthermore, when $B=V$, the above result yields an upper bound for the Laplacian eigenvalues of graphs, improving the previously known bounds of Kelner, Lee, Price and Teng [Geom. Funct. Anal. 21 (2011), 1117--1143] and Amini and Cohen-Steiner [Comment. Math. Helv. 93 (2018), 203--223].
Comments19 pages, 3 figures
