De Bruijn digraphs and affine transformations

Let Z d n be the additive group of 1 × n row vectors over Z d . For an n × n matrix T over  Z d and ω ∈ Z d n , the affine transformation F T , ω of Z d n sends x to x T + ω . Let 〈 α 〉 be the cyclic group generated by a vector α ∈ Z d n . The affine transformation coset pseudo-digraph TCP ( Z d n , α , F T , ω ) has the set of cosets of 〈 α 〉 in Z d n as vertices and there are c arcs from x + 〈 α 〉 to y + 〈 α 〉 if and only if the number of z ∈ x + 〈 α 〉 such that F T , ω ( z ) ∈ y + 〈 α 〉 is c . We prove that the following statements are equivalent: (a)  TCP ( Z d n , α , F T , ω ) is isomorphic to the d -nary ( n − 1 ) -dimensional De Bruijn digraph; (b)  α is a cyclic vector for T ; (c)  TCP ( Z d n , α , F T , ω ) is primitive. This strengthens a result conjectured by C.M. Fiduccia and E.M. Jacobson [Universal multistage networks via linear permutations, in: Proceedings of the 1991 ACM/IEEE Conference on Supercomputing, ACM Press, New York, 1991, pp. 380–389]. Under the further assumption that T is invertible we show that each component of TCP ( Z d n , α , F T , ω ) is a conjunction of a cycle and a De Bruijn digraph, namely a generalized wrapped butterfly. Finally, we discuss the affine TCP digraph representations for a class of digraphs introduced by D. Coudert, A. Ferreira and S. Perennes [Isomorphisms of the De Bruijn digraph and free-space optical networks, Networks 40 (2002) 155–164].