Eigenvalues and extremal degrees of graphs

Let G be a graph with n   vertices, μ1(G)⩾⋯⩾μn(G)μ1(G)⩾⋯⩾μn(G) be the eigenvalues of its adjacency matrix, and 0=λ1(G)⩽⋯⩽λn(G)0=λ1(G)⩽⋯⩽λn(G) be the eigenvalues of its Laplacian. We show thatδ(G)⩽μk(G)+λk(G)⩽Δ(G)for all1⩽k⩽nandμk(G)+μn-k+2(G¯)⩾δ(G)-Δ(G)-1for all2⩽k⩽n.Let GG be an infinite family of graphs. We prove that GG is quasi-random if and only if μn(G)+μn(G¯)=o(n) for every G∈GG∈G of order n  . This also implies that if λn(G)+λn(G¯)=n+o(n) (or equivalently λ2(G)+λ2(G¯)=o(n)) for every G∈GG∈G of order n  , then GG is quasi-random.