The exponent of the primitive Cayley digraphs on finite Abelian groups Original Research Article

Let G be finite group and let S be a subset of G. We prove a necessary and sufficient condition for the Cayley digraph X(G, S) to be primitive when S contains the central elements of G. As an immediate consequence we obtain that a Cayley digraph X(G, S) on an Abelian group is primitive if and only if S−1S is a generating set for G. Moreover, it is shown that if a Cayley digraph X(G, S) on an Abelian group is primitive, then its exponent either is n − l, [n2], [n2] − 1 or is not exceeding [n3] + 1. Finally, we also characterize those Cayley digraphs on Abelian groups with exponent n − 1, [n2], [n2] − 1. In particular, we generalize a number of well-known results for the primitive circulant matrices.