Spin Models on Bipartite Distance-Regular Graphs

Spin models were introduced by V. Jones (Pac. J. Math.137 (1989), 311-336) to construct invariants of knots and links. A spin model will be defined as a pair S = (X, w) of a finite set X and a function w on X × X satisfying several axioms. Some important spin models can be constructed on a distance-regular graph Γ = (X, E) with suitable complex numbers t0, t1, ..., td (d is the diameter of Γ) by putting w(a, b) = t∂(a, b). In this paper we determine bipartite distance-regular graphs which give spin models in this way with distinct t1, ..., td We show that such a bipartite distance-regular graph satisfies a strong regularity condition (it is 2-homogeneous), and we classify bipartite distance-regular graphs which satisfy this regularity condition.