A new inequality for distance-regular graphs
type: Item
instance of: scholarly article
title: A new inequality for distance-regular graphs
zbMATH Open document ID: 0814.05074
DOI: 10.1016/0012-365X(93)E0170-9
published in: Discrete Mathematics
publication date: 1995-06-08T00:00:00Z
review text: Let \(\Gamma\) be a distance-regular graph with \(n\) vertices and diameter \(d\geq 3\). Let \(i\) be fixed and \(d(x,y)= i\). If \(d(x, z)= 1\) then the possible values for \(d(y, z)\) are \(i\), \(i+1\), \(i-1\). The intersection numbers \(a_ i\), \(b_ i\), \(c_ i\) denote the corresponding number of possible vertices \(z\) and are independent of the choice of \(x\) and \(y\). Let \(A\) be an \(n\times n\) distance matrix and \(M\) a commutative (semi- simple) Bose-Mesner algebra, i.e. \(\mathbb{R}\)-algebra, generated by \(A\). Let \(E\) be a primitive idempotent of \(M\), \(AE= EA= \theta E\) and \(nE= \sum^ d_ 0\theta^*_ k A^ k\). An inequality, involving intersection numbers, \(\theta \leq \theta^*_ k\) is proved for each \(3\leq i\leq d\). It appears that equality is attained for \(i= 3\) if and only if it is attained for all \(i\) and this happens if and only if \(\Gamma\) is a \(Q\)-polynomials with respect to \(E\). The inequalities looks simpler if for some \(q\) values \(qc_ i- b_ i q(qc_{i-1}- b_{i-1})\) are independent of \(i\). For \(q\neq 0, 1, -1\) they can be writen as \(c_ i\frac {q^{2-i}- 1}{1- q^ i}\geq c_{i- 1}(q^{2- i}- 1)\).
Mathematics Subject Classification ID: 05E30
zbMATH DE Number: 719470
zbMATH Keywords: association scheme
zbMATH Keywords: distance-regular graph
zbMATH Keywords: intersection numbers
zbMATH Keywords: distance matrix
zbMATH Keywords: Bose-Mesner algebra
zbMATH Keywords: primitive idempotent
zbMATH Keywords: inequality
zbMATH Keywords: \(Q\)-polynomials
author: Paul Terwilliger
reviewed by: Victor Ufnarovski
MaRDI profile type: Publication
cites work: Q3218140
cites work: The graphs with spectral radius between 2 and \(\sqrt{2+\sqrt{5}}\)
cites work: Q4194971
cites work: Krein conditions and near polygons
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full work available at URL: 0012-365x(93)e0170-9
OpenAlex ID: W2020693441
description: scientific article; zbMATH DE number 719470