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

Abstract

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.