Eigenvalue comparisons in graph theory

Abstract

Let Γ be a finite graph with degree bounded below by k. Let λ1, λ2, . . . , λN denote the eigenvalues of the adjacency operator on Γ, arranged in non-increasing order. We derive lower bounds for the first several λi in terms of k and the diameter of Γ. Our bounds arise from a study of the roots of spherical eigenfunctions of the adjacency operator on a k-tree. We transplant these eigenfunctions onto Γ to construct test functions whose Rayleigh quotients are easy to estimate.