Extension and rigidity of Perrin's lower bound estimate for Steklov eigenvalues on graphs

TL;DR

The paper extends Perrin's lower bound for Steklov eigenvalues from unit-weight graphs to general weighted graphs and provides a rigidity description of the equality case. It develops the Steklov framework on weighted graphs with boundary, derives a strengthened bound $\\sigma_2 \\ge \\frac{w_0 V_B}{(V_B - m_0)^2 d_B}$, and characterizes when equality occurs, showing the graph must be a comb over a boundary path with specific mass and edge-weight conditions. The results unify and strengthen Perrin's estimates in the discrete setting and reveal a precise combinatorial structure (a comb) underpinning optimality. This advances discrete geometric analysis of Steklov problems and has potential implications for isoperimetric-type and spectral questions on graphs.

Abstract

In this paper, we extend a lower bound estimate for Steklov eigenvalues by Perrin \cite{Pe} on unit-weighted graphs to general weighted graphs and characterise its rigidity.

Theorems & Definitions (8)

• Theorem 1.1: Perrin Pe
• Theorem 1.2: Perrin Pe
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Definition 2.2
• proof : Proof of Theorem \ref{['thm-Pe-g']}
• proof : Proof of Theorem \ref{['thm-Pe-r']}