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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Vydané v:	Electronic notes in discrete mathematics Ročník 54; s. 157 - 162
Hlavní autori:	Fàbrega, J., Martí-Farré, J., Muñoz, X.
Médium:	Journal Article Publikácia
Jazyk:	English
Vydavateľské údaje:	Elsevier B.V 01.10.2016
Predmet:	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
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Abstract	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.
AbstractList	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 $S_{i}^star(v)$ be the set of vertices at distance $i$ from $v$. We show that $\|S_{i}^star(v)\|=d^i-a_{i-1}d^{i-1}-cdots -a_{1} d-a_{0}$, where $d$ is the degree of the digraph and the coefficients $a_{k}in{0,1}$ are explicitly calculated. Analogously, let $w$ be a vertex adjacent from $v$ such that $S_{i}^star(v)cap S_j^{ast}(w)neq emptyset$ for some $j$. We prove that $big \|S_{i}^star(v) cap S_j^{ast}(w) big \|=d^i-b_{i-1}d^{i-1}-ldots -b_{1} d-b_{0},$ where the coefficients $b_{t}in{0,1}$ are determined from the coefficients $a_k$ of the polynomial expression of $\|S_{i}^star(v)\|$. 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. Peer Reviewed 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.
Author	Muñoz, X. Martí-Farré, J. Fàbrega, J.
Author_xml	– sequence: 1 givenname: J. surname: Fàbrega fullname: Fàbrega, J. email: josep.fabrega@upc.edu – sequence: 2 givenname: J. surname: Martí-Farré fullname: Martí-Farré, J. email: jaume.marti@upc.edu – sequence: 3 givenname: X. surname: Muñoz fullname: Muñoz, X. email: xavier.munoz@upc.edu
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SubjectTerms	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
Title	Layer structure of De Bruijn and Kautz digraphs. An application to deflection routing
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