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Homogeneity of a Distance-Regular Graph Which Supports a Spin Model

Authors:	Brian Curtin Kazumasa Nomura

Institution:	(1) Department of Mathematics, University of South Florida, 4202 E. Fowler Ave, PHY 114, Tampa, FL 33647, USA;(2) College of Liberal Arts and Sciences, Tokyo Medical and Dental University, Kohnodai, Ichikawa 272-0827, Japan

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 whose Bose-Mesner algebra $\mathcal{M}$ satisfies W $\mathcal{M}$ $\mathcal{N}$ (W). Suppose W has at least three distinct entries. We show that is 1-homogeneous and that the first and the last subconstituents of are strongly regular and distance-regular, respectively.

Keywords:	distance-regular graph 1-homogeneous spin model
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