Highly arc transitive diagraphs
European Journal of Combinatorics
Universal multistage networks via linear permutations
Proceedings of the 1991 ACM/IEEE conference on Supercomputing
Universality of iterated networks
SPAA '92 Proceedings of the fourth annual ACM symposium on Parallel algorithms and architectures
Cayley Digraphs Based on the de Bruijn Networks
SIAM Journal on Discrete Mathematics
Discrete Applied Mathematics
Cayley digraphs form complete generalized cycles
European Journal of Combinatorics
Shift Register Sequences
Generalized de Bruijn digraphs and the distribution of patterns in α-expansions
Discrete Mathematics
Some characterizations for the wrapped butterfly
Analysis, combinatorics and computing
Hi-index | 0.00 |
Let Zdn be the additive group of 1 × n row vectors over Zd. For an n × n matrix T over Zd and ω ∈ Zdn, the affine transformation FT,ω of Zdn sends x to xT + ω. Let 〈α〉 be the cyclic group generated by a vector α ∈ Zdn. The affine transformation coset pseudo-digraph TCP(Zdn, α, FT,ω) has the set of cosets of 〈α〉 in Zdn 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(Zdn, α, FT,ω) is isomorphic to the d-nary (n - 1)-dimensional De Bruijn digraph; (b) α is a cyclic vector for T; (c) TCP(Zdn, α, FT,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(Zdn, 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 D. Coudert, A. Ferreira and S. Perennes [Isomorphisms of the De Bruijn digraph and free-space optical networks, Networks 40 (2002) 155-164].