Spectra and eigenspaces from regular partitions of Cayley (di)graphs of permutation groups
In this paper, we present a method to obtain regular (or equitable) partitions of Cayley (di)graphs (that is, graphs, digraphs, or mixed graphs) of permutation groups on n letters. We prove that every partition of the number n gives rise to a regular partition of the Cayley graph. By using represent...
| Autores: | , |
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| Formato: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2020 |
| País: | España |
| Recursos: | Universitat de Lleida (UdL) |
| Repositorio: | Repositori Obert UdL |
| OAI Identifier: | oai:repositori.udl.cat:10459.1/68317 |
| Acesso em linha: | https://doi.org/10.1016/j.laa.2020.03.015 http://hdl.handle.net/10459.1/68317 |
| Access Level: | acceso abierto |
| Palavra-chave: | Lifted (di)graph Regular partition Spectrum |
| Resumo: | In this paper, we present a method to obtain regular (or equitable) partitions of Cayley (di)graphs (that is, graphs, digraphs, or mixed graphs) of permutation groups on n letters. We prove that every partition of the number n gives rise to a regular partition of the Cayley graph. By using representation theory, we also obtain the complete spectra and the eigenspaces of the corresponding quotient (di)graphs. More precisely, we provide a method to find all the eigenvalues and eigenvectors of such (di)graphs, based on their irreducible representations. As examples, we apply this method to the pancake graphs P(n) and to a recent known family of mixed graphs Γ(d, n, r) (having edges with and without direction). As a byproduct, the existence of perfect codes s in P(n) allows us to give a lower bound for the multiplicity of its eigenvalue −1. |
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