  In our recent 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 $d$.
