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Home > Vol 21, No 3 (2005)

John S. Caughman¹ and Nadine Wolff²

¹Department of Mathematical Sciences Portland State University P.O. Box 751 Portland OR 97207-0751
²Department of Mathematics University of Hawaii at Hilo 200 W. Kawili St. Hilo HI 96720

Let denote a distance-regular graph with vertex set X, diameter D 3, valency k 3, and assume supports a spin model W. Write W = _{i = 0} ^D t _i A _i where A _i is the ith distance-matrix of . To avoid degenerate situations we assume is not a Hamming graph and t _i { t ₀, - t ₀ } for 1 i D. In an earlier paper Curtin and Nomura determined the intersection numbers of in terms of D and two complex parameters and q. We extend their results as follows. Fix any vertex x X and let T = T( x) denote the corresponding Terwilliger algebra. Let U denote an irreducible T-module with endpoint r and diameter d. We obtain the intersection numbers c _i( U), b _i( U), a _i( U) as rational expressions involving r, d, D, and q. We show that the isomorphism class of U as a T-module is determined by r and d. We present a recurrence that gives the multiplicities with which the irreducible T-modules appear in the standard module. We compute these multiplicites explicitly for the irreducible T-modules with endpoint at most 3. We prove that the parameter q is real and we show that if is not bipartite, then q > 0 and is real.

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