The Faber-Krahn type isoperimetric inequalities for a graph

Abstract

In this paper, a graph theoretic analog to the celebrated Faber-Krahn inequality for the first eigenvalue of the Dirichlet problem of the Laplacian for a bounded domain in the Euclidean space is shown. Namely, the optimal estimate of the first eigenvalue of the Dirichlet boundary problem of the combinatorial Laplacian for a graph with boundary is given.