On sum of powers of the Laplacian eigenvalues of graphs

Let G=(V,E)G=(V,E) be a simple graph with vertex set V(G)={v1,v2,…,vn}V(G)={v1,v2,…,vn} and edge set E(G)E(G). The Laplacian matrix of G   is L(G)=D(G)−A(G)L(G)=D(G)−A(G), where D(G)D(G) is the diagonal matrix of its vertex degrees and A(G)A(G) is the adjacency matrix. Let μ1⩾μ2⩾⋯⩾μn−1⩾μn=0μ1⩾μ2⩾⋯⩾μn−1⩾μn=0 be the Laplacian eigenvalues of G. For a graph G   and a real number β≠0β≠0, the graph invariant Sβ(G)Sβ(G) is the sum of the β-th power of the non-zero Laplacian eigenvalues of G, that is,Sβ(G)=∑i=1n−1μiβ.In this paper, we obtain some lower and upper bounds on Sβ(G)Sβ(G) for G in terms of n, the number of edges m  , maximum degree Δ1Δ1, clique number ω, independence number α and the number of spanning trees t  . Moreover, we present some Nordhaus–Gaddum-type results for Sβ(G)Sβ(G) of G.