Note on "Average resistance of toroidal graphs" by Rossi, Frasca and Fagnani

Abstract: In our paper W.S. Rossi, P. Frasca and F. Fagnani, "Average resistance of toroidal graphs", SIAM Journal on Control and Optimization, 53(4):2541--2557, 2015, we studied how the average resistances of $d$-dimensional toroidal grids depend on the graph topology and on the dimension of the graph. Our results were based on the connection between resistance and Laplacian eigenvalues. In this note, we contextualize our work in the body of literature about random walks on graphs. Indeed, the average effective resistance of the $d$-dimensional toroidal grid is proportional to the mean hitting time of the simple random walk on that grid. If $d\geq3 $, then the average resistance can be bounded uniformly in the number of nodes and its value is of order $1/d$ for large $d$.