On the structure of graph edge designs that optimize the algebraic connectivity

— We take a structural approach to the problem of designing the edge weights in an undirected graph subject to an upper bound on their total, so as to maximize the algebraic connectivity. Specifically, we first characterize the eigenvector(s) associated with the algebraic connectivity at the optimum, using optimization machinery together with eigenvalue sensitivity notions. Using these characterizations, we obtain an alternative finite-search algorithm for finding the optimal design in tree graphs that is quadratic in the number of vertices, and further address update of the design upon addition of a new vertex. We also obtain a suite of results concerning the topological and eigen-structure of optimal designs for bipartite and general graphs. In turn, we obtain a lower-bound on the optimal algebraic connectivity in terms of the graph’s diameter, and also describe how our structural insights can inform and be meshed with numerical solution techniques. Finally, an example concer...

• Type: Conference
• Year: 2008
• Where: CDC
• Authors: Yan Wan, Sandip Roy, Xu Wang, Ali Saberi, Tao Yang, Mengran Xue, Babak Malek
• Keywords: Algebraic Connectivity, CDC 2008, Control Systems, Optimal Algebraic Connectivity, Optimal Designs