Abstract
In this paper we apply Polya's Theorem to the problem of enumerating Cayley graphs on permutation groups up to isomorphisms induced by conjugacy in the symmetric group. We report the results of a search of all three-regular Cayley graphs on permutation groups of degree at most nine for small diameter graphs. We explore several methods of constructing covering graphs of these Cayley graphs. Examples of large graphs with small diameter are obtained.
Highlights
The (∆, D) problem asks for the largest value n such that a graph on n vertices exists with diameter D and maximum vertex degree ∆
The Moore bound for the diameter D of a graph with n vertices and maximum vertex degree ∆ ≥ 3 gives n ≤ 1 + ∆ + (∆ − 1)∆ + (∆ − 1)2∆ + · · · + (∆ − 1)D−1∆
Jerrum and Skyum [JS84] have obtained the best non-random constructive results known for an infinite family of cubic graphs
Summary
Discrete Mathematics and Theoretical Computer Science, DMTCS, 2001, 4 (2), pp.123132. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés
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