The smallest eigenvalue of Kr-free graphs


Let G be a Kr+1-free graph with n vertices and m edges, and let λn(G) be the smallest eigenvalue of its adjacency matrix. We show thatλn(G)<-2r+1mr/rn2r-1.This implies also that if G is a d-regular graph of order n and independence number r, the second eigenvalue of G satisfiesλ2(G)≥-1+2/r(n-1-d)r/n r-1. © 2006 Elsevier B.V. All rights reserved.

Publication Title

Discrete Mathematics