Effective resistance of toroidal graphs; some sharper results and applications

The average effective resistance of a graph can be computed as a function of its eigenvalues. Using this formula and focusing on toroidal d-dimensional graphs, we study the role of the network topology and specifically of the graph dimension in determining the resistance. Considering sequences of graphs of increasing size, we study the asymptotical behavior of the effective resistance, proving that it is (asymptotically) inversely proportional to the dimension. These findings are relevant in many applications, including distributed estimation and control of sensor networks and clock networks.