Steklov flows on trees and applications

2021/03/13 by Zunwu He, Bobo Hua, He, Zunwu +1 · 3 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2103.07696

Abstract

We introduce the Steklov flows on finite trees, i.e. the flows (or currents) associated with the Steklov problem. By constructing appropriate Steklov flows, we prove the monotonicity and rigidity of the first nonzero Steklov eigenvalues on trees: for finite trees \g₁ and \g₂, the first nonzero Steklov eigenvalue of \g₁ is greater than or equal to that of \g₂, provided that \g₁ is a subgraph of \g₂. Moreover, we give the sufficient and necessary condition in which the equality holds.

Cited by

• Sharp bounds and monotonicity results for Neumann eigenvalues
• A generalized Cheeger inequality and the Steklov Problem on finite graphs
• Comparison between the first Steklov eigenvalue and algebraic connectivity on trees

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