Layer structure of De Bruijn and Kautz digraphs. An application to deflection routing - Fàbrega, J.; Martí-Farré, J.; Muñoz, X. (Electronic notes in discrete mathematics, 2016)
In the main part of this paper we present polynomial expressions for the cardinalities of some sets of interest of the nice distance-layer structure of the well-known De Bruijn and Kautz digraphs. More precisely, given a vertex v, let Si⋆ (v) be the set of vertices at distance i from v. We show that |Si⋆(v)|=di−ai−1di−1−⋯−a1d−a0, where d is the degree of the digraph and the coefficients ak∈{0,1} are explicitly calculated. Analogously, let w be a vertex adjacent from v such that Si⋆(v)∩Sj⁎(w)≠∅ for some j. We prove that |Si⋆(v)∩Sj⁎(w)|=di−bi−1di−1−…−b1d−b0, where the coefficients bt∈{0,1} are determined from the coefficients ak of the polynomial expression of |Si⋆(v)|. An application to deflection routing in De Bruijn and Kautz networks serves as motivation for our study. It is worth-mentioning that our analysis can be extended to other families of digraphs on alphabet or to general iterated line digraphs.