Designing node and edge weights of a graph to meet Laplacian eigenvalue constraints

We consider agents connected over a network, and propose a method to design an optimal interconnection such that the gap between the largest and smallest Laplacian eigenvalues of the graph representing the network is minimized. We study ways of imposing constraints that may arise in physical systems, such as enforcing lower bounds on connectivity and upper bounds on gain as well as network cost. In particular, we show that node and edge weights of a given graph can be simultaneously adjusted via convex optimization to achieve improvements in its Laplacian spectrum.