Counting pure k cycles in sequences of Cayley graphs Original Research Article

For each positive integer n we consider sequences Xn of Cayley graph Cay(Gn, Sn), where Gn is a finite abelian group and Sn is a symmetric subset of Gn. The sequence Xn has the multiplicative arithmetic property (map) if for each pair of positive relatively prime integers (m, n) there is a group isomorphism ϕn, m from Gnm to Gn × Gm such that ϕn, m maps Snm onto Sn × Sm. Let Xn have the map and let pk(n) denote the number of induced k cycles of Xn. We show that 2k pk is a linear combination (over Z) of multiplicative arithmetic functions. In particular, the sequence Cay(Zn, Un), where Zn is the ring of integers modulo n and Zn is the multiplicative group of units modulo n has the map. For this sequence explicit formulae for p3(n) and p4(n) in terms of the primes dividing n are given.