Tails of bipartite distance-regular graphs
type: Item
instance of: scholarly article
title: Tails of bipartite distance-regular graphs
zbMATH Open document ID: 1012.05159
author: Michael S. Lang
published in: European Journal of Combinatorics
publication date: 2003-03-18T00:00:00Z
review text: Let \(\Gamma \) denote a bipartite distance-regular graph with diameter \(D\geq 4\) and valency \(k\geq 3\). Let \(\theta _{0}>\theta _{1}>\cdots >\theta _{D}\) denote the eigenvalues of \(\Gamma \) and let \(E_{0}, E_{1},\ldots ,E_{D}\) denote the associated primitive idempotents. Fix \(s\) \((1\leq s\leq D-1)\); \(E_{s}\) is said to be a tail whenever the entrywise product \(E_{s}\circ E_{s}\) is a linear combination of \(E_{0}\), \(E_{s}\) and at most one other primitive idempotent of \(\Gamma \). Let \(q_{ij}^{h}\) \((0\leq h,i,j\leq D)\) denote the Krein parameters of \(\Gamma \) and let \(\Delta \) denote the undirected graph with vertices \(0,1,\ldots ,D\) where two vertices \(i,j\) are adjacent whenever \(i\neq j\) and \(q_{ij}^{s}\neq 0\). The main result of this paper asserts that \(E_{s}\) is a tail if and only if one of (i)--(iii) holds: (i) \(\Delta \) is a path; (ii) \(\Delta \) has two connected components, each of them is a path, one consisting of the even vertices and one consisting of the odd vertices; (iii) \(D=6\) and \(\Delta \) has two connected components, one is a path on four even vertices and the other is a clique on three odd vertices.
Mathematics Subject Classification ID: 05E30
zbMATH DE Number: 1884183
zbMATH Keywords: distance-regular graph
zbMATH Keywords: eigenvalue
zbMATH Keywords: primitive idempotent
zbMATH Keywords: Krein parameters
zbMATH Keywords: diameter
reviewed by: Ioan Tomescu
MaRDI profile type: Publication
full work available at URL: eujc.2002.0611
OpenAlex ID: W2060267832
cites work: Q3992965
cites work: A new inequality for bipartite distance-regular graphs.
cites work: A new inequality for distance-regular graphs
DOI: 10.1006/EUJC.2002.0611
description: scientific article; zbMATH DE number 1884183