Higher order Dirichlet-to-Neumann maps on graphs and their eigenvalues

TL;DR

This paper introduces higher order Dirichlet-to-Neumann maps on graphs, extending previous work and providing eigenvalue estimates, thus bridging discrete graph theory with concepts from Riemannian geometry.

Contribution

It defines higher order DtN maps on graphs and derives eigenvalue estimates, generalizing prior models and connecting discrete and continuous geometric analysis.

Findings

Derived eigenvalue estimates for the new DtN maps

Extended the concept of DtN maps to higher orders on graphs

Connected graph-based DtN maps with Riemannian manifold theory

Abstract

In this paper, we first introduce higher order Dirichlet-to-Neumann maps on graphs which can be viewed as a discrete analogue of the corresponding Dirichlet-to-Neumann maps on compact Riemannian manifolds with boundary and a higher order generalization of the Dirichlet-to-Neumann map on graphs introduced by Hua-Huang-Wang\cite{HHW} and Hassannezhad-Miclo \cite{HM}. Then, some Raulot-Savo-type estimates on the eigenvalues of the DtN maps introduced are derived.