Laplacian spectral moment and Laplacian Estrada index of random graphs

Let G be a simple graph with n vertices and let μ1≥μ2≥⋯≥μn=0 be the Laplacian eigenvalues of G. The k-th Laplacian spectral moment of G is defined to beLMk(G)=∑i=1nμik(G), where k is a non-negative integer; and the Laplacian Estrada index of G is defined asLEE(G)=∑i=1neμi. In this paper, we first estimate these two indices for almost all graphs, and then we give lower and upper bounds to these two indices for almost all multipartite graphs.