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

TL;DR

This paper extends the Faber-Krahn inequality to normalized combinatorial p-Laplacians on graphs, showing the tadpole graph uniquely minimizes the first Dirichlet eigenvalue.

Contribution

It provides a simpler proof that the tadpole graph uniquely minimizes the first eigenvalue for the normalized combinatorial p-Laplacian, extending prior inequalities.

Findings

Tadpole graph uniquely minimizes the first Dirichlet eigenvalue.

Extension of Katsuda-Urakawa's Faber-Krahn inequality to p-Laplacians.

Simpler proof method compared to previous work.

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.