Upper bounds of Steklov eigenvalues on graphs

Abstract

Let $\hat{I}”$ 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 $\ddot{I}{ƒ}_2$ of $G$ with boundary $B$. Using metrical deformation via flows, we first show that $\ddot{I}{ƒ}_2=\mathcal{O}\left(\frac{\hat{I}”(g+1{)}^3}{|B|}\right)$ for graphs of orientable genus $g$ if $|B|\ge \max \{3\sqrt{g},|V{|}^{\frac{1}{4}+\hat{I}\mu },9\}$ for some $\hat{I}\mu >0$. This can be seen as a discrete analogue of Karpukhin's bound. Secondly, we prove that $\ddot{I}{ƒ}_2\le \frac{8\hat{I}”+4X}{|B|}$ based on planar crossing number $X$. Thirdly, we show that $\ddot{I}{ƒ}_2\le \frac{|B|}{|B|-1}\cdot \hat{I}{´}_B$, where $\hat{I}{´}_B$ denotes the minimum degree for boundary vertices in $B$. At last, we compare several upper bounds on Laplacian eigenvalues and Steklov eigenvalues.

22 pages, 2 figures