Spin models and almost bipartite 2-homogeneous graphs. (English) Zbl 0858.05101

Bannai, E. (ed.) et al., Progress in algebraic combinatorics. Based on the international symposium on algebraic combinatorics, Fukuoka, Japan, 1993 and the research project on algebraic combinatorics, Kyoto, Japan, April 1994 – March 1995. Kyoto: Kinokuniya Company Ltd. Adv. Stud. Pure Math. 24, 285-308 (1996).
Motivated by spin models, a mathematical object introduced by V. Jones [Pac. J. Math. 137, No. 2, 311-324 (1989; Zbl 0695.46029)] to construct invariants of knots and links, the author studies and classifies 2-homogeneous almost bipartite distance regular graphs. Here a graph is called almost bipartite, if there is no cycle of odd length \(<2d+1\), where \(d\) is the diameter of the graph. In a first result, the author shows that if a spin model is constructed on an almost bipartite distance regular graph then the graph must be 2-homogeneous. A graph is called 2-homogeneous if for all vertices \(u\), \(x\), \(y\) with distances \(d(u,x)=d(u,y)\), \(d(x,y)=2\) the cardinality \(|\Gamma_{i-1}(u)\cap \Gamma_1(x)\cap \Gamma_1(y)|\) with \(i=d(u,y)\) is constant. Here for a vertex \(v\) the set \(\Gamma_j(v)\) is the set of all vertices having distance \(j\) from \(v\). Then in a second result the author classifies all 2-homogeneous almost bipartite distance regular graphs (i.e., all almost bipartite distance regular graphs corresponding to spin models).
For the entire collection see [Zbl 0845.00006].
Reviewer: V.Welker (Essen)


05E30 Association schemes, strongly regular graphs


Zbl 0695.46029