The smallest eigenvalue of KrKr-free graphs

Let G   be a Kr+1Kr+1-free graph with n vertices and m   edges, and let λn(G)λn(G) be the smallest eigenvalue of its adjacency matrix. We show thatλn(G)<-2r+1mrrn2r-1.This implies also that if G is a d-regular graph of order n   and independence number rr, the second eigenvalue of G satisfiesλ2(G)⩾-1+2r(n-1-d)rnr-1.