Cayley Digraphs from Complete Generalized Cycles

The complete generalized cycleG(d,n) is the digraph which hasZn × Zdas the vertex set and every vertex (i,x) is adjacent to thedvertices (i + 1,y) withy ∈ Zd. As a main result, we give a necessary and sufficient condition for the iterated line digraphG(d,n,k) = Lk − 1G(d,n), withda prime number, to be a Cayley digraph in terms of the existence of a groupΓdof orderdand a subgroupNof (Γd)nisomorphic to (Γd)k. The condition is shown to be also sufficient for any integerd ≥ 2. IfΓdis a ringRandNis a submodule ofRn, it is said thatG(d,n,k) is anR-Cayley digraph. By using some properties of the homogeneous linear recurrences in finite rings, necessary and sufficient conditions forG(d,n,k) to be anR-Cayley digraph are obtained. As a consequence, whenR = Zda new characterization for the digraphsG(d,n,k) to beZd-Cayley digraphs is derived. As a corollary, sufficient conditions for the corresponding underlying graphs to be Cayley can be deduced. Ifdis a prime power andFdis a finite field of orderd, the digraphsG(d,n,k) which areFd-Cayley digraphs are in 1-1 correspondence with the cyclic (n,k)-linear codes overFd.