Characterization of extremal graphs from Laplacian eigenvalues and the sum of powers of the Laplacian eigenvalues of graphs

For any real number α, let ^sα(G) denote the sum of the αth power of the non-zero Laplacian eigenvalues of a graph G. In this paper, we first obtain sharp bounds on the largest and the second smallest Laplacian eigenvalues of a graph, and a new spectral characterization of a graph from its Laplacian eigenvalues. Using these results, we then establish sharp bounds for ^sα(G) in terms of the number of vertices, number of edges, maximum vertex degree and minimum vertex degree of the graph G, from which a Nordhaus-Gaddum type result for ^sα is also deduced. Moreover, we characterize the graphs maximizing ^sα for α>1 among all the connected graphs with given matching number.