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An extension of Katsuda-Urakawa's Faber-Krahn inequality

Wankai He, Chengjie Yu

Abstract

In this paper, motivated by our previous work \cite{HY}, we prove that the minimum of the first Dirichlet eigenvalues for the normalized combinatorial $p$-Laplacian on connected finite graphs with boundary consisting of $n$ edges is only achieved by the tadpole graph $T_{n,3}$. This result extends the Faber-Krahn inequality of Katsuda-Urakawa \cite{KU} to normalized combinatorial $p$-Laplacians. Our argument is much simpler than that of Katsuda-Urakawa.

An extension of Katsuda-Urakawa's Faber-Krahn inequality

Abstract

In this paper, motivated by our previous work \cite{HY}, we prove that the minimum of the first Dirichlet eigenvalues for the normalized combinatorial -Laplacian on connected finite graphs with boundary consisting of edges is only achieved by the tadpole graph . This result extends the Faber-Krahn inequality of Katsuda-Urakawa \cite{KU} to normalized combinatorial -Laplacians. Our argument is much simpler than that of Katsuda-Urakawa.

Paper Structure

This paper contains 2 sections, 5 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Proof of the main result

Key Result

Theorem 1.1

Let $G$ be a connected graph with boundary that consists of $n\geq 4$ edges. Then, and the equality holds if and only if $G=T_{n,3}$.

Theorems & Definitions (6)

  • Theorem 1.1: Katsuda-Urakawa KU
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof