Taut distance-regular graphs of even diameter
type: Item
instance of: scholarly article
title: Taut distance-regular graphs of even diameter
zbMATH Open document ID: 1047.05044
author: Mark S. MacLean
published in: Journal of Combinatorial Theory. Series B
publication date: 2004-08-06T00:00:00Z
review text: Let \(\Gamma\) be a distance-regular graph with diameter \(D\geq 4\) and valency \(k\geq 3\). Let \(k=\theta_0>\cdots >\theta_D\) denote the distinct eigenvalues of \(\Gamma\), and for \(0\leq i\leq D\), let \(E_i\) denote the primitive idempotents of \(\Gamma\) associated with \(\theta_i\). A primitive idempotent \(F\) of \(\Gamma\) is called trivial, if \(F=E_0\) or \(\Gamma\) is bipartite and \(F=E_D\). Let \(E,F\) denote primitive idempotents of \(\Gamma\). The pair \(E,F\) is called taut whenever \(E,F\) are nontrivial and the entry-wise product \(E\circ F\) is a linear combination of two distinct primitive idempotents of \(\Gamma\). We define \(\Gamma\) to be taut, if \(\Gamma\) is not 2-homogeneous and \(\Gamma\) has the taut pair \(E,F\) of primitive idempotents. Theorem. Let \(\Gamma\) be a bipartite distance-regular graph with even diameter \(D\geq 4\), valency \(k\geq 3\), and eigenvalues \(\theta_0>\dots >\theta_D\). Let \(\theta\) denote an eigenvalue of \(\Gamma\) other than \(\theta_0,\theta_D\), and let \(\sigma_0,\dots ,\sigma_D\) denote the cosine sequence associated with \(\theta\). Then the following are equivalent: (i) \(\Gamma\) is taut or 2-homogeneous, and \(\theta\in \{\theta_1,\theta_{D-1}\}\). (ii) There exists a complex number \(\lambda\) such that \(\sigma_{i-1}-\lambda \sigma_i+\sigma_{i+1}=0\) for \(i\) odd, \(1\leq i\leq D-1\). (iii) There exists a complex number \(\lambda\) such that \(\sigma_{i-1}-\lambda \sigma_i+\sigma_{i+1}=0\) for \(i=1,3\).
Mathematics Subject Classification ID: 05E30
zbMATH DE Number: 2082036
zbMATH Keywords: distance-regular graph
zbMATH Keywords: bipartite graph
zbMATH Keywords: tight graph
zbMATH Keywords: taut graph
reviewed by: Alexandre A. Makhnev
MaRDI profile type: Publication
full work available at URL: j.jctb.2003.11.002
OpenAlex ID: W2046962284
cites work: Q3218140
cites work: Q3992965
cites work: 2-homogeneous bipartite distance-regular graphs
cites work: Dual bipartite Q-polynomial distance-regular graphs
cites work: Problems in algebraic combinatorics
cites work: Distance-regular graphs and halved graphs
cites work: 1-homogeneous graphs with cocktail party \({\mu}\)-graphs
cites work: A local approach to 1-homogeneous graphs
cites work: Nonexistence of some antipodal distance-regular graphs of diameter four
cites work: Krein parameters and antipodal tight graphs with diameter 3 and 4
cites work: Tight distance-regular graphs
cites work: An inequality involving two eigenvalues of a bipartite distance-regular graph
cites work: Taut distance-regular graphs of odd diameter
cites work: Homogeneous graphs and regular near polygons
cites work: Spin models on bipartite distance-regular graphs
cites work: An inequality in character algebras
cites work: Tight distance-regular graphs and the \(Q\)-polynomial property
cites work: Tight graphs and their primitive idempotents
cites work: An inequality on the cosines of a tight distance-regular graph
cites work: The subconstituent algebra of an association scheme. I
cites work: On the primitive idempotents of distance-regular graphs
DOI: 10.1016/J.JCTB.2003.11.002
description: scientific article; zbMATH DE number 2082036