  Discrete Green's functions are the inverses or pseudo-inverses of
combinatorial Laplacians. We present compact formulas for discrete Green's
functions, in terms of the eigensystems of corresponding Laplacians, for
products of regular graphs with or without boundary. Explicit formulas are
derived for the cycle, torus, and 3-dimensional torus, as is an inductive
formula for the $t$-dimensional torus with $n$ vertices, from which the Green's
function can be completely determined in time $O(t n^{2-1/t}\log{n})$. These
Green's functions may be used in conjunction with diffusion-like problems on
graphs such as electric potential, random walks, and chip-firing games or other
balancing games.
