Partially metric association schemes with a multiplicity three
van Dam,Edwin R. Koolen,Jack H. Park,J.
van Dam,Edwin R.
Koolen,Jack H.
Park,J.
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.
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Date
2018-05-01
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Research Projects
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Keywords
association scheme, 2-walk-regular graph, small multiplicity, distance-regular graph, cover of the cube
Citation
van Dam, E R, Koolen, J H & Park, J 2018, 'Partially metric association schemes with a multiplicity three', Journal of Combinatorial Theory Series B, vol. 130, pp. 19-48. https://doi.org/10.1016/j.jctb.2017.09.011