SQUARING GRAPHS MODULO n In a recent Combinatorics seminar, David A. Jackson introduced the following family of directed graphs

In a recent Combinatorics seminar, David A. Jackson introduced the following family of directed graphs {Γn}: Let n ∈ N. The vertex set is V (Γn) = Z/nZ. The directed edge (x, y) ∈ E(Γn) if and only if y = x. We allow loops and double edges. In particular, for any n, the edges (0, 0) and (1, 1) belong to E(Γn). As for double edges, (2, 4) and (4, 2) are both in E(Γ7). Γn consists of some number of connected components. In the seminar, it was proved that each component of Γn contains a unique cycle. Here, we count a loop as a 1-cycle, and a directed pair of edges as a 2-cycle. There is a directed tree attached to each vertex in a given cycle. These observations lead to a number of natural questions: (1) How many connected components does Γn have? (2) What cycle lengths appear in Γn? (3) With what frequency does a particular cycle length appear in Γn? (4) Can we describe the structure of the trees attached to the cycles in Γn? (5) Given k ∈ N, does there exist an n so that Γn has a k cycle? If so, what is the smallest such n? What is the structure of all such n? Our study attempts to provide satisfactory answers to these questions. The first three questions are very closely related, and we give relatively complete answers to these. In particular, if we could provide an answer to each of these whenever n is a prime power, then we would have a complete answer for general n. In addition, we do have some fairly explicit answers to these questions when p is a prime power. As for the last question, we are able to show that for any k, we can find infinitely many primes p so that Γp contains a cycle of length k. However, we do not know how to estimate the size of the smallest n for which Γn contains a k cycle.