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

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.