A sharp upper bound on the largest Laplacian eigenvalue of weighted graphs

We consider weighted graphs, where the edge weights are positive definite matrices. The Laplacian of the graph is defined in the usual way. We obtain an upper bound on the largest eigenvalue of the Laplacian and characterize graphs for which the bound is attained. The classical bound of Anderson and Morley, for the largest eigenvalue of the Laplacian of an unweighted graph follows as a special case.