Homotopy in \(Q\)-polynomial distance-regular graphs
type: Item
instance of: scholarly article
title: Homotopy in \(Q\)-polynomial distance-regular graphs
zbMATH Open document ID: 0972.05056
DOI: 10.1016/S0012-365X(00)00045-5
author: Heather A. Lewis
published in: Discrete Mathematics
publication date: 2000-12-03T00:00:00Z
review text: A connected graph \(\Gamma= (X,R)\) with diameter \(d\geq 1\) is called distance-regular if for all integers \(h\), \(i\), \(j\) such that \(0\leq h\), \(i,j\leq d\), and for all \(x,y\in X\) with \(\partial(x,y)= h\) the number \(p^h_{ij}= |\{z\in X\mid\partial(x, z)= i\text{ and }\partial(y,z)= j\}|\) depends only on \(h\), \(i\), \(j\) and not on \(x\) or \(y\). The integers \(p^h_{ij}\) are called the intersection numbers of \(\Gamma\). Since in a distance-regular graph \(\Gamma\) we have \(p^0_{11}= k\), \(\Gamma\) must be regular with valency \(k\). Considering the concept of the Bose-Mesner algebra of a distance-regular graph \(\Gamma\) we say that \(\Gamma\) is \(Q\)-polynomial if \(\Gamma\) has at least one \(Q\)-idempotent. The author proves that if \(\Gamma= (X,R)\) is a \(Q\)-polynomial distance-regular graph with diameter \(d\geq 3\) and valency \(k\geq 3\), then the intersection number \(p^3_{12}\) is at least 2 and the girth of \(\Gamma\) is at most 6. The author also defines the concept of a pseudoquotient graph and proves that if \(\Gamma\) is not a pseudoquotient graph, then any cycle in \(\Gamma\) can be decomposed into cycles of length at most 6.
reviewed by: M. R. Darafsheh
Mathematics Subject Classification ID: 05E30
zbMATH DE Number: 1538206
zbMATH Keywords: homotopy
zbMATH Keywords: intersection numbers
zbMATH Keywords: distance-regular graph
zbMATH Keywords: Bose-Mesner algebra
zbMATH Keywords: \(Q\)-polynomial
zbMATH Keywords: cycle
MaRDI profile type: Publication
Wikidata QID: Q128109353
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description: scientific article; zbMATH DE number 1538206