Eigenvalues and forbidden subgraphs I

Suppose a graph G have n vertices, m edges, and t triangles. Letting λn(G) be the largest eigenvalue of the Laplacian of G and μn(G) be the smallest eigenvalue of its adjacency matrix, we prove that(λn (G) ≥ frac(2 m2 - 3 nt, m (n2 - 2 m)) n,; μn (G) ≤ frac(3 n3 t - 4 m3, nm (n2 - 2 m)),)with equality if and only if G is a regular complete multipartite graph. Moreover, if G is Kr+1-free, thenλn (G) ≥ frac(2 mn, (r - 1) (n2 - 2 m))with equality if and only if G is a regular complete r-partite graph. © 2006 Elsevier Inc. All rights reserved