On the multiplicity of eigenvalues of distance-regular graphs
type: Item
instance of: scholarly article
title: On the multiplicity of eigenvalues of distance-regular graphs
zbMATH Open document ID: 0832.05075
DOI: 10.1016/0024-3795(95)00152-H
author: Jack H. Koolen
author: Chris Godsil
published in: Linear Algebra and its Applications
publication date: 1996-02-20T00:00:00Z
review text: The main goal of this note is to give better lower bounds on the multiplicity of eigenvalues in a distance-regular graph. The main result is Theorem 4: Let \(\Gamma\) be a distance-regular graph with \(a_d= 0\). Let \(\Theta\) be an eigenvalue of \(\Gamma\) with multiplicity \(f\). Suppose that the standard array belonging to \(\Theta\) satisfies \(u_2\neq u_0\neq u_d\). Then \(f> k+ b_{d- 1}- 1\) or \(\Theta\in \{\Theta_1, \Theta_d\}\) and either \(\Theta+ 1\) is an integer dividing \(b_1\) or \(\Theta_1+ 1\) and \(\Theta_d+ 1\) are conjugate algebraic integers dividing \(b_1\). This rules out the infinite series of feasible intersection arrays \[ \{\mu(2\mu+ 1), (\mu- 1)(2\mu+ 1), \mu^2, \mu; 1, \mu, \mu(\mu- 1), \mu(2\mu+ 1): \mu\geq 2\}. \]
Mathematics Subject Classification ID: 05C50
zbMATH DE Number: 803742
zbMATH Keywords: multiplicity of eigenvalues
zbMATH Keywords: distance-regular graph
zbMATH Keywords: standard array
zbMATH Keywords: feasible intersection arrays
reviewed by: Jack E. Graver
MaRDI profile type: Publication
cites work: Q3992965
cites work: Problems in algebraic combinatorics
cites work: Distance regular covers of the complete graph
cites work: A new feasibility condition for distance-regular graphs
description: scientific article; zbMATH DE number 803742