### Abstract

Original language | English |
---|---|

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 |

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### Keywords

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

### Cite this

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

*130*, 19-48. https://doi.org/10.1016/j.jctb.2017.09.011

}

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

**Partially metric association schemes with a multiplicity three.** / van Dam, Edwin; Koolen, Jack H.; Park, J.

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - Partially metric association schemes with a multiplicity three

AU - van Dam, Edwin

AU - Koolen, Jack H.

AU - Park, J.

PY - 2018/5/1

Y1 - 2018/5/1

N2 - 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.

AB - 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.

KW - association scheme

KW - 2-walk-regular graph

KW - small multiplicity

KW - distance-regular graph

KW - cover of the cube

U2 - 10.1016/j.jctb.2017.09.011

DO - 10.1016/j.jctb.2017.09.011

M3 - Article

VL - 130

SP - 19

EP - 48

JO - Journal of Combinatorial Theory, Series B, Graph theory

JF - Journal of Combinatorial Theory, Series B, Graph theory

SN - 0095-8956

ER -