Some Formulas for Spin Models on Distance-Regular Graphs

A spin model is a square matrixWsatisfying certain conditions which ensure that it yields an invariant of knots and links via a statistical mechanical construction of V. F. R. Jones. Recently F. Jaeger gave a topological construction for each spin modelWof an association scheme which containsWin its Bose–Mesner algebra. Shortly thereafter, K. Nomura gave a simple algebraic construction of such a Bose–Mesner algebraN(W). In this paper we study the caseW∈A⊆N(W), where A is the Bose–Mesner algebra of a distance-regular graph. We show the following results. LetΓ=(X, R) be a distance-regular graph of diameterd>1 such that the Bose–Mesner algebra A ofΓsatisfiesW∈A⊆N(W) for some spin modelWonX. WriteW=∑di=0 tiAi, whereAidenotes theith adjacency matrix. Setxi=t−1i−1tiandp=x−11x2. Thenxi=pi−1x1holds for alli. Moreover, the eigenvalues and the intersection numbers ofΓare rational functions ofx1andp.