  We apply eigenvalue interlacing techniques for obtaining lower and upper
bounds for the sums of Laplacian eigenvalues of graphs, and characterize
equality. This leads to generalizations of, and variations on theorems by
Grone, and Grone and Merris. As a consequence we obtain inequalities involving
bounds for some well-known parameters of a graph, such as edge-connectivity,
and the isoperimetric number.
