Eigenvalue estimates for the fractional Laplacian on lattice subgraphs

TL;DR

This paper extends classical eigenvalue estimates to the fractional Laplacian on lattice subgraphs, providing bounds for the sum of the first eigenvalues under Dirichlet conditions.

Contribution

It introduces the fractional Laplacian on subgraphs with Dirichlet boundary conditions and establishes eigenvalue bounds for lattice graphs, extending classical results.

Findings

Derived upper and lower bounds for eigenvalues on lattice subgraphs.

Extended Li-Yau and Kröger eigenvalue estimates to fractional Laplacians.

Provided a framework for analyzing fractional operators on discrete structures.

Abstract

We introduce the the fractional Laplacian on a subgraph of a graph with Dirichlet boundary condition. For a lattice graph, we prove the upper and lower estimates for the sum of the first $k$ Dirichlet eigenvalues of the fractional Laplacian, extending the classical results by Li-Yau and Kr\"{o}ger.