On Laplacian Eigenvalues of a Graph

Let G be a connected graph with n vertices and m edges. The Laplacian eigenvalues are denoted by μ1(G) ≥ μ2(G)≥ · · · ≥μn−1(G) > μn(G) = 0. The Laplacian eigenvalues have important applications in theoretical chemistry. We present upper bounds for μ1(G)+· · ·+μk(G) and lower bounds for μn−1(G)+· · ·+μn−k(G) in terms of n and m, where 1 ≤ k ≤ n−2, and characterize the extremal cases. We also discuss a type of upper bounds for μ1(G) in terms of degree and 2-degree.