### Abstract

An association scheme is called partially metric if it has a connected relation whose distance-two relation is also a relation of the scheme. In this paper we determine the symmetric partially metric association schemes with a multiplicity three. Besides the association schemes related to regular complete 4-partite graphs, we obtain the association schemes related to the Platonic solids, the bipartite double scheme of the dodecahedron, and three association schemes that are related to well-known 2-arc-transitive covers of the cube: the Möbius–Kantor graph, the Nauru graph, and the Foster graph F048A. In order to obtain this result, we also determine the symmetric association schemes with a multiplicity three and a connected relation with valency three. Moreover, we construct an infinite family of cubic arc-transitive 2-walk-regular graphs with an eigenvalue with multiplicity three that give rise to non-commutative association schemes with a symmetric relation of valency three and an eigenvalue with multiplicity three.

Original language | English |
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Pages (from-to) | 19-48 |

Journal | Journal of Combinatorial Theory, Series B, Graph theory |

Volume | 130 |

Early online date | Oct 2017 |

DOIs | |

Publication status | Published - 1 May 2018 |

### Keywords

- association scheme
- 2-walk-regular graph
- small multiplicity
- distance-regular graph
- cover of the cube

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## Cite this

van Dam, E., Koolen, J. H., & Park, J. (2018). Partially metric association schemes with a multiplicity three.

*Journal of Combinatorial Theory, Series B, Graph theory*,*130*, 19-48. https://doi.org/10.1016/j.jctb.2017.09.011