Faber-Krahn inequalities for first Dirichlet eigenvalues of combinatorial $p$-Laplacian on graphs with boundary

TL;DR

This paper establishes sharp Faber-Krahn inequalities for the first Dirichlet eigenvalues of the combinatorial p-Laplacian on graphs, identifying the tadpole graph as the minimizer for fixed vertices or edges.

Contribution

It provides the first sharp inequalities for these eigenvalues on graphs and characterizes the extremal graph as the tadpole graph for p>1.

Findings

Minimum eigenvalues achieved only on the tadpole graph T_{n,3}

Sharp inequalities for the first Dirichlet eigenvalues

Extends Faber-Krahn inequalities to combinatorial p-Laplacian on graphs

Abstract

In this paper, we obtain sharp Faber-Krahn inequalities for the first Dirichlet eigenvalue of the combinatorial $p$-Laplacian on connected graphs with a fixed number of vertices or with a fixed number of edges. More precisely, we show that the minimum of the first $p$-Dirichlet eigenvalues of connected graphs with boundary that consist of $n$ vertices or $n$ edges is achieved only on the tadpole graph $T_{n,3}$ when $p>1$.