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 $S_{i}^\star(v)$ be the set of vertices at distance $i$ from $v...
| Authors: | , , |
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| Format: | article |
| Publication Date: | 2016 |
| Country: | España |
| Institution: | Universitat Politècnica de Catalunya (UPC) |
| Repository: | UPCommons. Portal del coneixement obert de la UPC |
| Language: | English |
| OAI Identifier: | oai:upcommons.upc.edu:2117/103040 |
| Online Access: | https://hdl.handle.net/2117/103040 https://dx.doi.org/10.1016/j.endm.2016.09.028 |
| Access Level: | Open access |
| Keyword: | Polynomials Matrices De Bruijn and Kautz digraphs General iterated line digraphs Deflection routing Polinomis Matrius (Matemàtica) Classificació AMS::11 Number theory::11C Polynomials and matrices Classificació AMS::12 Field theory and polynomials::12Y05 Computational aspects of field theory and polynomials Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de nombres Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de cossos i polinomis |
| 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 $S_{i}^\star(v)$ 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. |
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