Abstract
A spin model is a square matrix that encodes the basic data for a statistical mechanical construction of link invariants due to V.F.R. Jones. Every spin model W is contained in a canonical Bose-Mesner algebra \(\mathcal{N}\)(W). In this paper we study the distance-regular graphs Γ whose Bose-Mesner algebra \(\mathcal{M}\) satisfies W ∈ \(\mathcal{M}\) ⊂ \(\mathcal{N}\)(W). Suppose W has at least three distinct entries. We show that Γ is 1-homogeneous and that the first and the last subconstituents of Γ are strongly regular and distance-regular, respectively.
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Curtin, B., Nomura, K. Homogeneity of a Distance-Regular Graph Which Supports a Spin Model. J Algebr Comb 19, 257–272 (2004). https://doi.org/10.1023/B:JACO.0000030702.58352.f7
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DOI: https://doi.org/10.1023/B:JACO.0000030702.58352.f7