New cyclic Kautz digraphs with optimal diameter
We obtain a new family of digraphs with minimal diameter, that is, given the number of vertices and out-degree, there is no other digraph with a smaller diameter. This new family of digraphs are called `modified cyclic digraphs' MCK(d,ℓ)MCK(d,ℓ), and it is derived from the Kautz digraphs K(d,ℓ)...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2021 |
| País: | España |
| Institución: | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Repositorio: | Recercat. Dipósit de la Recerca de Catalunya |
| OAI Identifier: | oai:recercat.cat:10459.1/83184 |
| Acceso en línea: | https://doi.org/10.11575/cdm.v16i3.62468 http://hdl.handle.net/10459.1/83184 |
| Access Level: | acceso abierto |
| Palabra clave: | Line digraph Diameter Kautz digraph |
| Sumario: | We obtain a new family of digraphs with minimal diameter, that is, given the number of vertices and out-degree, there is no other digraph with a smaller diameter. This new family of digraphs are called `modified cyclic digraphs' MCK(d,ℓ)MCK(d,ℓ), and it is derived from the Kautz digraphs K(d,ℓ)K(d,ℓ) and from the so-called cyclic Kautz digraphs CK(d,ℓ)CK(d,ℓ). The cyclic Kautz digraphs CK(d,ℓ)CK(d,ℓ) were defined as the digraphs whose vertices are labeled by all possible sequences a1…aℓa1…aℓ of length ℓℓ, such that each character aiai is chosen from an alphabet of d+1d+1 distinct symbols, where the consecutive characters in the sequence are different (as in Kautz digraphs), and also requiring that a1≠aℓa1≠aℓ. Their arcs are between vertices a1a2…aℓa1a2…aℓ and a2…aℓaℓ+1a2…aℓaℓ+1, with a1≠aℓa1≠aℓ and a2≠aℓ+1a2≠aℓ+1. Since CK(d,ℓ)CK(d,ℓ) do not have minimal diameter for their number of vertices, we construct the modified cyclic Kautz digraphs to obtain the same diameter as in the Kautz digraphs, and we also show that MCK(d,ℓ)MCK(d,ℓ) are dd-out-regular. Moreover, for t≥1t≥1, we compute the number of vertices of the iterated line digraphs Lt(CK(d,ℓ))Lt(CK(d,ℓ)). |
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