On Laplacian eigenvalues of a graph

Let G be a connected graph with n vertices and in edges. The Laplacian eigenvalues are denoted by mu(1) (G) greater than or equal to mu(2) (G) greater than or equal to(...)greater than or equal to mu(n-1) (G) > mu(n) (G) = 0. The Laplacian eigenvalues have important applications in theoretical chemistry. We present upper bounds for mu(1) (G) + (...) + mu(k) (G) and lower bounds for mu(n-1) (G) + (...) + mu(n-k) (G) in terms of n and m, where 1 less than or equal to k less than or equal to n-2, and characterize the extremal cases. We also discuss a type of upper bounds for mu(1) (G) in terms of degree and 2-degree.