Spin Models on Triangle-Free Connected Graphs

Spin models were introduced by V. Jones (Pac. J. Math.137(1989), 311–334) to construct invariants of knots and links. A spin model is defined as a pairS=(X, w) of a fine setXand a functionw: X×X→Csatisfying several axioms. LetΛ=(X, E) be a connected graph with the usual metric ∂: X×X→{0, 1, …, d}, whereddenotes the diameter ofΛ. It is shown that, ifΛhas no 3-cycle, and ifS=(X, t∘∂) is a spin model for a mappingt: {0, 1, …, d}→Csatisfying some conditions (which hold if t is injective), thenΛis an almost bipartite distance-regular graph.