De Bruijn digraphs and affine transformations

Let Zd be the additive group of 1×n row vectors over Zd. For an n×nmatrix T over Zd and ω ∈ Z n d , the affine transformation FT,ω of Z n d sends x to xT +ω. Let 〈α〉 be the cyclic group generated by a vector α ∈ Zd . The affine transformation coset pseudo-digraph TCP (Zd , α, FT,ω) has the set of cosets of 〈α〉 in Z n d 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 FT,ω(z) ∈ y+ 〈α〉 is c. We prove that the following statements are equivalent: (a) TCP (Zd , α, FT,ω) is isomorphic to the d-nary (n − 1)-dimensional De Bruijn digraph; (b) TCP (Zd , α, FT,ω) is primitive; (c) α is a cyclic vector for T . This strengthens a result conjectured by Fiduccia and Jacobson [Universal multistage networks via linear permutations, in: Proceedings of the 1991 ACM/IEEE Conference on Supercomputing, ACM Press, 1991, New York, pp. 380– 389]. Under the further assumption that T is invertible we show that each component of TCP (Zd , α, FT,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 Coudert, Ferreira and Perennes [Isomorphisms of the De Bruijn digraph and free-space optical networks, Networks 40 (2002) 155–164]. Keywords– affine transformation, De Bruijn digraph, wrapped butterfly, transformation coset pseudo-digraph.