Eigenvalue Comparison Theorems of Neumann Laplacian for Graphs

Abstract

In this paper, the spectrum of the Neumann Laplacian for a graph with boundary is studied. Two comparison theorems of the Neumann Laplacian for a graph are shown. Namely, the optimal upper and lower bounds of the first eigenvalue of the Neumann boundary problem of the combinatorial Laplacian for a graph with boundary are given.