Spin models on triangle-free connected graphs

Spin models were introduced by V. Jones (Pnc. J. Math. 137 (1989), 311-334) to construct invariants of knots and links. A spin model is defined as a pair S = (X, w) of a fine set X and a function w: X x X --> C satisfying several axioms. Let Gamma = (X, E) be a connected graph with the usual metric partial derivative: X x X --> {0, 1,..., d}, where d denotes the diameter of Gamma. It is shown that, if Gamma has no 3-cycle, and if S = (X, t circle partial derivative) is a spin model for a mapping t: {0, 1,..., d} --> C satisfying some conditions (which hold if 1 is injective), then Gamma is an almost bipartite distance-regular graph. (C) 1996 Academic Press, Inc.