Tight graphs and their primitive idempotents
type: Item
instance of: scholarly article
title: Tight graphs and their primitive idempotents
zbMATH Open document ID: 0927.05085
DOI: 10.1023/A:1018624103247
author: Arlene A. Pascasio
published in: Journal of Algebraic Combinatorics
publication date: 1999-12-05T00:00:00Z
review text: Let \(\Gamma\) be a distance-regular graph with diameter \(d\geq 3\) and eigenvalues \(\theta_0>\theta_1>\dots>\theta_d\). Then \[ \Biggl( \theta_1+\frac{k}{a_1+1}\Biggr) \Biggl(\theta_d+\frac{k}{a_1+1}\Biggr)\geq \frac{-ka_1b_1}{(a_1+1)^2}. \] \(\Gamma\) is said to be tight whenever \(\Gamma\) is not bipartite and equality holds above. Suppose \(E\) and \(F\) are nontrivial primitive idempotents of \(\Gamma\) and the entry-wise product \(E\circ F\) is a scalar multiple of a primitive idempotent \(H\) of \(\Gamma\). Then \(\Gamma\) is either bipartite and at least one of \(E,\;F\) is equal to \(E_d\) or tight and \(\{E,F\}=\{E_1,E_d\}\) (in last case the eigenvalue associated with \(H\) is \(\theta_{d-1}\) and \(k\theta_{d-1}= \theta_1\theta_d\)).
Mathematics Subject Classification ID: 05E30
zbMATH DE Number: 1319552
zbMATH Keywords: tight graph
zbMATH Keywords: primitive idempotent
zbMATH Keywords: Krein parameter
zbMATH Keywords: distance regular graph
zbMATH Keywords: eigenvalues
reviewed by: Alexandre A. Makhnev
MaRDI profile type: Publication
cites work: Q3218140
cites work: Q3992965
cites work: Problems in algebraic combinatorics
cites work: Tight distance-regular graphs
description: scientific article; zbMATH DE number 1319552