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

For any real number αα, let sα(G)sα(G) denote the sum of the ααth power of the non-zero Laplacian eigenvalues of a graph GG. 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)sα(G) in terms of the number of vertices, number of edges, maximum vertex degree and minimum vertex degree of the graph GG, from which a Nordhaus–Gaddum type result for sαsα is also deduced. Moreover, we characterize the graphs maximizing sαsα for α>1α>1 among all the connected graphs with given matching number.