Classification of Partially Metric Q-Polynomial Association Schemes with $$m_{ 1} = 4$$
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ORIGINAL PAPER
Classification of Partially Metric Q-Polynomial Association Schemes with m1 = 4 Da Zhao1 Received: 15 October 2019 / Revised: 9 September 2020 / Accepted: 25 September 2020 Ó Springer Japan KK, part of Springer Nature 2020
Abstract We classify the Q-polynomial association schemes with m1 ¼ 4 which are partially metric with respect to the nearest neighbourhood relation. An association scheme is partially metric with respect to a relation R1 if the scheme graph of R2 is exactly the distance-2 graph of the scheme graph of R1 under a certain ordering of the relations. Keywords Association scheme Spherical embedding 4 dimensional Euclidean geometry Regular polytope
Mathematics Subject Classification Primary 05E30 Secondary 52C99
1 Background The classification problem plays an import role in the research of association scheme. Hanaki and Miyamoto give the list of association schemes up to 30 vertices [11]. There are studies on association schemes of small degree, including [14, 22]. Bannai and Bannai finished the classification of primitive association schemes with multiplicity three [2]. The Terwilliger algebra arises from the classification project of P and Q-polynomial association schemes. The classification of P-polynomial association schemes (distance-regular graphs) of valency three or four is done in [6, 8]. The Bannai-Ito conjecture and its dual, proved by Bang–Dubickas–Koolen– Moulton [1] and Martin–Williford [17], guarantee the finiteness of P-polynomial (Q-polynomial) association schemes of given valency (multiplicity). However, the classification of Q-polynomial association schemes of small multiplicity is still Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00373020-02237-x) contains supplementary material, which is available to authorized users. & Da Zhao [email protected] 1
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China
123
Graphs and Combinatorics
unfinished. Bannai and the author revisited the paper [2] and obtained the classification of Q-polynomial association schemes with m1 ¼ 3 in [5]. In this paper, we aim to finish the classification of Q-polynomial association schemes with m1 ¼ 4. The paper is organized as follows. In Sect. 2, we introduce association schemes and related concepts. In Sect. 3, we state our main result. The tools we used are exhibited in Sect. 4 and the proof of the main theorem is given in Sect. 5.
2 Preliminaries 2.1 Graphs
V A (simple) graph C consists of vertices V and edges E . Every graph in this 2 paper is simple unless specified otherwise. For a graph C, its complementary graph C is the graph with the same vertex set as C and with an edge uv in C if and only if uv is not an edge in C. A walk on the graph C (from vertex x to vertex y of length ‘) is a sequence of vertices x ¼ v0 ; v1 ; . . .v‘ ¼ y such that vi1 vi is an edge for i ¼ 1; 2; . . .; ‘. The distance dist ðx; yÞ between two vertices x and y is the le
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