Eigenvalues and forbidden subgraphs I Original Research Article

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 thatimagewith equality if and only if G is a regular complete multipartite graph. Moreover, if G is Kr+1-free, thenimagewith equality if and only if G is a regular complete r-partite graph.