Cycles of directed graphs defined by matrix multiplication (mod n)

Let A be a k × k matrix over a ring R; let GM (A, R) be the digraph with vertex set R k , and an arc from v to w if and only if w = Av. In this paper, we determine the numbers and lengths of the cycles of GM (A, R) for k = 2 in the following two cases. (a) R = F F q , the q–element finite field, and (b) R = Z Z/nZ Z and GCD(n, det(A)) = 1. This extends previous results for k = 1 and R = Z Z/nZ Z. We make considerable use of the Smith form of a matrix; other than that, the most powerful tool we use is the Chinese Remainder Theorem.