Spin models on triangle-free connected graphs
Spin models were introduced by \textit{V. Jones} [Pac. J. Math. 137, No. 2, 311-334 (1989; Zbl 0695.46029)] to construct invariants of knots and links. A spin model is defined as a pair \(S=(X,w)\) of a finite set \(X\) and a function \(w:X\times X\to \mathbb{C}\) satisfying several axioms. Let \(\Gamma=(X,E)\) be a connected graph with the usual metric \(\partial:X\times X\to \{0,1,\dots,d\}\), where \(d\) denotes the diameter of \(\Gamma\). It is shown that, if \(\Gamma\) has no 3-cycle, and if \(S=(X,t\circ\partial)\) is a spin model for a mapping \(t:\{0,1,\dots,d\}\to\mathbb{C}\) satisfying some conditions (which hold if \(t\) is injective), then \(\Gamma\) is an almost bipartite distance-regular graph.
scientific article; zbMATH DE number 934816