A digraph is said to be a c-circulant if its adjacency matrix is c-circulant. This paper deals with general properties of this family of digraphs, as isomorphisms, regularity, strong connectivity, diameter and the relation between c-circulant digraphs and the line digraph technique.
Mora, M., Serra, O., Fiol, M. General properties of c-circulant digraphs. "Ars combinatoria", 1988, vol. 25C, p. 241-252.