Upper bounds for the sum of Laplacian eigenvalues of graphs

Let G be a graph with n vertices and e(G) edges, and let μ1(G)⩾μ2(G)⩾⋯⩾μn(G)=0 be the Laplacian eigenvalues of G. Let , where . Brouwer conjectured that for . It has been shown in Haemers et al. [7] that the conjecture is true for trees. We give upper bounds for Sk(G), and in particular, we show that the conjecture is true for unicyclic and bicyclic graphs.