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

Wankai He, Chengjie Yu

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 $p>1$.

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

Abstract

In this paper, we obtain sharp Faber-Krahn inequalities for the first Dirichlet eigenvalue of the combinatorial -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 -Dirichlet eigenvalues of connected graphs with boundary that consist of vertices or edges is achieved only on the tadpole graph when .
Paper Structure (3 sections, 7 theorems, 89 equations)

This paper contains 3 sections, 7 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs of main results

Key Result

Theorem 1.1

Let $G$ be a connected graph with boundary. Suppose that $|E(G)|=n\geq 4$. Then, and the equality holds if and only if $G=T_{n,3}$.

Theorems & Definitions (14)

  • Theorem 1.1: Katsuda-Urakawa KU
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 4 more