Tight Estimates for Eigenvalues of Regular Graphs

It is shown that if a d-regular graph contains s vertices so that the distance between any pair is at least 4k, then its adjacency matrix has at least s eigenvalues which are at least 2 d - 1 cos( 2k ). A similar result has been proved by Friedman using more sophisticated tools. 1 The main result Let G = (V, E) be a simple d-regular graph on n vertices. let A be its adjacency matrix, and let 1(G) = d 2(G) . . . n(G) be its eigenvalues. Alon and Boppana ([1], see also [9], [11]) proved that for any fixed d and for any infinite family of of d-regular graphs Gi, lim inf 2(Gi) 2

• Type: Journal
• Year: 2004
• Where: COMBINATORICS
• Authors: Alon Nilli
• Keywords: Adjacency Matrix, COMBINATORICS 2004, D-regular Graphs, Simple D-regular Graph