Eigenvalues and extremal degrees of graphs

Let G be a graph with n vertices, μ1(G) μn(G) be the eigenvalues of its adjacency matrix, and 0=λ1(G) λn(G) be the eigenvalues of its Laplacian. We show that and Let be an infinite family of graphs. We prove that is quasi-random if and only if for every of order n. This also implies that if (or equivalently ) for every of order n, then is quasi-random.