Moore mixed graphs from Cayley graphs
A Moore (r, z, k)-mixed graph G has every vertex with undirected degree r, directed in- and outdegree z, diameter k, and number of vertices (or order) attaining the corresponding Moore bound M(r, z, k) for mixed graphs. When the order of G is close to M(r, z, k) vertices, we refer to it as an almost...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2023 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/387468 |
| Acceso en línea: | https://hdl.handle.net/2117/387468 https://dx.doi.org/10.5614/ejgta.2023.11.1.15 |
| Access Level: | acceso abierto |
| Palabra clave: | Graph theory Group theory Mixed graph Moore bound Cayley graph Line digraph Spectrum Grafs, Teoria de Grups, Teoria de Classificació AMS::05 Combinatorics::05C Graph theory Classificació AMS::20 Group theory and generalizations::20C Representation theory of groups Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de grups |
| Sumario: | A Moore (r, z, k)-mixed graph G has every vertex with undirected degree r, directed in- and outdegree z, diameter k, and number of vertices (or order) attaining the corresponding Moore bound M(r, z, k) for mixed graphs. When the order of G is close to M(r, z, k) vertices, we refer to it as an almost Moore graph. The first part of this paper is a survey about known Moore (and almost Moore) general mixed graphs that turn out to be Cayley graphs. Then, in the second part of the paper, we give new results on the bipartite case. First, we show that Moore bipartite mixed graphs with diameter three are distance-regular, and their spectra are fully characterized. In particular, an infinity family of Moore bipartite (1, z, 3)-mixed graphs is presented, which are Cayley graphs of semidirect products of groups. Our study is based on the line digraph technique, and on some results about when the line digraph of a Cayley digraph is again a Cayley digraph. |
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