Layer structure of De Bruijn and Kautz digraphs. An application to deflection routing

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...

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Bibliographic Details
Published in:Electronic notes in discrete mathematics Vol. 54; pp. 157 - 162
Main Authors: Fàbrega, J., Martí-Farré, J., Muñoz, X.
Format: Journal Article Publication
Language:English
Published: Elsevier B.V 01.10.2016
Subjects:
11 Number theory
11C Polynomials and matrices
12 Field theory and polynomials
12Y05 Computational aspects of field theory and polynomials
Classificació AMS
De Bruijn and Kautz digraphs
Deflection routing
General iterated line digraphs
Matemàtiques i estadística
Matrices
Matrius (Matemàtica)
Polinomis
Polynomials
Teoria de cossos i polinomis
Teoria de nombres
Àlgebra
Àrees temàtiques de la UPC
General iterated line digraphs
Deflection routing
De Bruijn and Kautz digraphs
ISSN:1571-0653, 1571-0653
Online Access:Get full text
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Summary: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.
ISSN:1571-0653
1571-0653
DOI:10.1016/j.endm.2016.09.028