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 thatλn(G)⩾2m2-3ntm(n2-2m)n,μn(G)⩽3n3t-4m3nm(n2-2m),with equality if and only if G is a regular complete multipartite graph.Moreover, if G is Kr+1-free, thenλn(G)⩾2mn(r-1)(n2-2m)with equality if and only if G is a regular complete r-partite graph.