Eigenvalue comparisons in graph theory

Gregory T. Quenell

Vol. 176 (1996), No. 2, 443–461

DOI: 10.2140/pjm.1996.176.443

Abstract

Let Γ be a finite graph with degree bounded below by k. Let λ₁,λ₂,…,λ_N denote the eigenvalues of the adjacency operator on Γ, arranged in non-increasing order. We derive lower bounds for the first several λ_i 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.

Mathematical Subject Classification 2000

Primary: 05C50

Milestones

Received: 22 August 1994

Published: 1 December 1996