Bipartite \(Q\)-polynomial distance-regular graphs
type: Item
instance of: scholarly article
title: Bipartite \(Q\)-polynomial distance-regular graphs
zbMATH Open document ID: 1054.05101
published in: Graphs and Combinatorics
publication date: 2004-12-13T00:00:00Z
review text: Let \(\Gamma\) denote a bipartite \(Q\)-polynomial distance-regular graph with diameter \(D\geq 4\). Then the intersection numbers of \(\Gamma\) are determined by \(D\) and two real scalars \(q\) and \(s^*\). It is proved that \(s^*=0\) if \(D\geq 12\). Theorem 1.1. Let \(\Gamma\) be a bipartite distance-regular graph with diameter \(D\geq 12\). Then \(\Gamma\) is \(Q\)-polynomial if and only if the following conditions (i)--(iv) hold: (i) \(\Gamma\) is the ordinary \(2D\)-cycle. (ii) \(\Gamma\) is the Hamming cube \(H(D,2)\). (iii) \(\Gamma\) is the antipodal quotient of \(H(2D,2)\). (iv) The intersection numbers of \(\Gamma\) satisfy \[ c_i=\frac{q^i-1}{q-1},\;b_i=\frac{q^D-q^i}{q-1},\;(0\leq i\leq D), \] where \(q\) is an integer not less than 2. The intersection numbers given in the case (iv) above are realized by both the bipartite dual polar graphs and the Hemmeter graph.
Mathematics Subject Classification ID: 05E30
zbMATH DE Number: 2121770
zbMATH Keywords: distance-regular graphs
zbMATH Keywords: Terwilliger algebra
author: John S. IV Caughman
reviewed by: Alexandre A. Makhnev
MaRDI profile type: Publication
full work available at URL: s00373-003-0538-8
OpenAlex ID: W2011493409
DOI: 10.1007/S00373-003-0538-8
description: scientific article; zbMATH DE number 2121770