Spin models on triangle-free connected graphs
type: Item
instance of: scholarly article
title: Spin models on triangle-free connected graphs
zbMATH Open document ID: 0857.05095
published in: Journal of Combinatorial Theory. Series B
publication date: 1997-03-06T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 05E30
zbMATH DE Number: 934816
zbMATH Keywords: knots
zbMATH Keywords: links
zbMATH Keywords: spin model
zbMATH Keywords: connected graph
zbMATH Keywords: diameter
zbMATH Keywords: distance-regular graph
author: Kazumasa Nomura
MaRDI profile type: Publication
OpenAlex ID: W2008894377
DOI: 10.1006/JCTB.1996.0046
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description: scientific article; zbMATH DE number 934816