Upper bounds of Steklov eigenvalues on graphs

Abstract

Let $Δ$ 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 $σ_2$ of $G$ with boundary $B$. Using metrical deformation via flows, we first show that $σ_2 = \mathcal{O}\left(\frac{Δ(g+1)^3}{|B|}\right)$ for graphs of orientable genus $g$ if $|B| \geq \max\{3 \sqrt{g},|V|^{\frac{1}{4} + ε}, 9\}$ for some $ε> 0$. This can be seen as a discrete analogue of Karpukhin's bound. Secondly, we prove that $σ_2 \leq \frac{8Δ+4X}{|B|}$ based on planar crossing number $X$. Thirdly, we show that $σ_2 \leq \frac{|B|}{|B|-1} \cdot δ_B$, where $δ_B$ denotes the minimum degree for boundary vertices in $B$. At last, we compare several upper bounds on Laplacian eigenvalues and Steklov eigenvalues.

Figures (2)

• Figure 1: Crossing of paths near a vertex in $G$
• Figure 2: A graph with boundary $(G, B)$ of maximum degree $\Delta$ such that $\sigma_2 = \Delta - \frac{4}{5}$.

Theorems & Definitions (44)

• Theorem 1.1: lin_first_2024
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Definition 2.1: Vertex congestion
• Definition 2.2: Integral boundary flow and unit $H$-boundary flow
• Lemma 2.3
• proof