There are finitely many triangle-free distance-regular graphs with degree 8, 9 or 10
type: Item
instance of: scholarly article
title: There are finitely many triangle-free distance-regular graphs with degree 8, 9 or 10
zbMATH Open document ID: 1047.05043
DOI: 10.1023/B:JACO.0000023006.04375.60
author: Jack H. Koolen
author: Vincent Moulton
published in: Journal of Algebraic Combinatorics
publication date: 2004-08-06T00:00:00Z
review text: Bannai and Ito conjectured that there are finitely many distance-regular graphs with degree at least 3. This conjecture holds for degrees 3 and 4 (Bannai, Ito), 5, 6 and 7 (Koolen, Moulton). Let \(\Gamma\) be a distance-regular graph with valency \(k\geq 3\) and diameter \(d\geq 2\). Given integers \(a\geq 0\), \(b,c\geq 1\) and \(a+b+c=k\), we define \(l_{(c,a,b)}=\{i\;|\;1\leq i\leq d-1,\;(c_i,a_i,b_i)=(c,a,b)\}\), and put \(h_\Gamma=l_{(1,a_1,b_1)}\) and \(t_\Gamma=l_{(b_1,a_1,1)}\). Theorem. Suppose \(k\geq 3\) is an integer. Then there exists a real number \(\alpha>0\), depending only on \(k\) so that there are finitely many triangle-free distance-regular graphs with degree \(k\) and diameter \(d\) satisfying \(d-(h_\Gamma+t_\Gamma)\leq \alpha h_\Gamma\). Corollary 1.5. There are finitely many triangle-free distance-regular graphs with degree 8, 9, or 10.
Mathematics Subject Classification ID: 05E30
zbMATH DE Number: 2082004
zbMATH Keywords: distance-regular graph
zbMATH Keywords: eigenvalues
zbMATH Keywords: Bannai-Ito conjecture
reviewed by: Alexandre A. Makhnev
MaRDI profile type: Publication
cites work: Q3832610
cites work: On distance-regular graphs with fixed valency
cites work: On distance-regular graphs with fixed valency. II
cites work: On distance-regular graphs with fixed valency. III
cites work: On distance-regular graphs with fixed valency. IV
cites work: Q3992965
cites work: On a conjecture of Bannai and Ito: There are finitely many distance-regular graphs with degree 5, 6 or 7
cites work: Eigenvalue multiplicities of highly symmetric graphs
Recommended article: Triangle-free distance-regular graphs
Recommended article: A note on triangle-free distance-regular graphs with \(a_2\neq 0\)
Recommended article: On distance-regular graphs \(\Gamma\) of diameter 3 for which \(\Gamma_3\) is a triangle-free graph
Recommended article: Triangle- and pentagon-free distance-regular graphs with an eigenvalue multiplicity equal to the valency
Recommended article: Triangle-free distance-regular graphs with an eigenvalue multiplicity equal to their valency and diameter 3
Recommended article: There are only finitely many distance-regular graphs of fixed valency greater than two
Recommended article: Triangle-Free Graphs with Large Degree
Recommended article: On some distance-regular graphs with many vertices
Recommended article: Distance-regular graphs of diameter 3 without triangles with \(c_2=2\)
Recommended article: There are only finitely many regular near polygons and geodetic distance-regular graphs with fixed valency
description: scientific article; zbMATH DE number 2082004