Some Formulas for Spin Models on Distance-Regular Graphs

Abstract

A spin model is a square matrix W satisfying 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 model W of an association scheme which contains W in its Bose–Mesner algebra. Shortly thereafter, K. Nomura gave a simple algebraic construction of such a Bose–Mesner algebra N ( W ). In this paper we study the case W ∈ A ⊆ N ( W ), where A is the Bose–Mesner algebra of a distance-regular graph. We show the following results. Let Γ =( X , R ) be a distance-regular graph of diameter d >1 such that the Bose–Mesner algebra A of Γ satisfies W ∈ A ⊆ N ( W ) for some spin model W on X . Write W =∑ d i =0 t i A i , where A i denotes the i th adjacency matrix. Set x i = t −1 i −1 t i and p = x −1 1 x 2 . Then x i = p i −1 x 1 holds for all i . Moreover, the eigenvalues and the intersection numbers of Γ are rational functions of x 1 and p .