Counting pure \(k\)-cycles in sequences of Cayley graphs (Q1910564)

For each positive integer \(n\), let \(G_n\) be a finite abelian group and \(S_n\) a symmetric subset of \(G_n\), i.e. \(S_n= - S_n\). \(\text{Cay}(G_n, S_n)\) is the Cayley graph of \(G_n\) with respect to \(S_n\). In this paper, sequences \(X_n\) of Cayley graphs \(\text{Cay}(G_n, S_n)\) are considered. The sequence \(X_n\) has the multiplicative arithmetic property if for each pair \((m, n)\) of positive integers with \(\text{gcd}(m, n)= 1\) there is a group isomorphism \(\varphi_{m, n}\) from \(G_{nm}\) to \(G_n\times G_m\) such that \(\varphi_{m, n}\) maps \(S_{nm}\) onto \(S_n\times S_m\). For a sequence \(X_n\) of Cayley graphs, \(p_k(n)\) denotes the number of induced \(k\)-cycles of \(X_n\). The main result is the following theorem: If a sequence \(X_n\) has the multiplicative arithmetic property, then the function \(2kp_k\) is a linear combination (with integer coefficients) of multiplicative arithmetic functions.