Cayley Digraphs from Complete Generalized Cycles

The complete generalized cycle G(d, n) is the digraph which has Zn × Zd as the vertex set and every vertex (i, x) is adjacent to the d vertices (i + 1, y) with y ∈ Zd . As a main result, we give a necessary and sufficient condition for the iterated line digraph G(d, n, k) = Lk−1G(d, n), with d a prime number, to be a Cayley digraph in terms of the existence of a group0d of order d and a subgroup N of (0d ) n isomorphic to (0d ) k . The condition is shown to be also sufficient for any integer d ≥ 2. If 0d is a ring R and N is a submodule of R n , it is said that G(d, n, k) is an R-Cayley digraph. By using some properties of the homogeneous linear recurrences in finite rings, necessary and sufficient conditions for G(d, n, k) to be an R-Cayley digraph are obtained. As a consequence, when R = Zd a new characterization for the digraphs G(d, n, k) to be Zd -Cayley digraphs is derived. As a corollary, sufficient conditions for the corresponding underlying graphs to be Cayley can be deduced. If d is a prime power and Fd is a finite field of order d , the digraphs G(d, n, k) which are Fd -Cayley digraphs are in 1-1 correspondence with the cyclic (n, k)-linear codes over Fd .