Discrete Green's functions for products of regular graphs

Abstract

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.