Layer structure of De Bruijn and Kautz digraphs: an application to deflection routing
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hdl:2117/103040
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Data publicació2016-10-17
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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 $S_{i}^\star(v)$ 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.
CitacióFàbrega, J., Martí-Farré, J., Muñoz, X. Layer structure of De Bruijn and Kautz digraphs: an application to deflection routing. "Electronic notes in discrete mathematics", 17 Octubre 2016, vol. 54, p. 157-162.
ISSN1571-0653
Versió de l'editorhttp://www.sciencedirect.com/science/article/pii/S1571065316301226
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