Partially metric association schemes with a multiplicity three

Edwin van Dam, Jack H. Koolen, J. Park

Research output: Contribution to journalArticleScientificpeer-review

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 languageEnglish
Pages (from-to)19-48
JournalJournal of Combinatorial Theory, Series B, Graph theory
Volume130
Early online dateOct 2017
DOIs
Publication statusPublished - 1 May 2018

Fingerprint

Association Scheme
Multiplicity
Metric
Graph in graph theory
Arc of a curve
Dodecahedron
Platonic solid
Eigenvalue
Regular Graph
Walk
Regular hexahedron
Cover

Keywords

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

Cite this

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title = "Partially metric association schemes with a multiplicity three",
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 M{\"o}bius–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.",
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Partially metric association schemes with a multiplicity three. / van Dam, Edwin; Koolen, Jack H.; Park, J.

In: Journal of Combinatorial Theory, Series B, Graph theory, Vol. 130, 01.05.2018, p. 19-48.

Research output: Contribution to journalArticleScientificpeer-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

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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

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