Spin models on bipartite distance-regular graphs. (English) Zbl 0827.05060

Nomura, K.

J. Comb. Theory, Ser. B 64, No. 2, 300-313 (1995).

Spin models were introduced by 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 \(w\) on \(X\times X\) satisfying several axioms. Some important spin models can be constructed on a distance-regular graph \(\Gamma= (X, E)\) with suitable complex numbers \(t_0, t_1, \dots, t_d\) (\(d\) is the diameter of \(\Gamma\)) by putting \(w(a,b)= t_{\partial (a,b)}\). In this paper we determine bipartite distance- regular graphs which give spin models in this way with distinct \(t_1, \dots, t_d\). 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.

Reviewer: K.Nomura (Tokyo)

MSC:

05E30

Association schemes, strongly regular graphs

Keywords:

state model; spin model; distance-regular graph; regularity condition