A quasi-spectral characterization of strongly distance-regular graphs
type: Item
instance of: scholarly article
title: A quasi-spectral characterization of strongly distance-regular graphs
zbMATH Open document ID: 0956.05103
author: R. Smith
published in: The Electronic Journal of Combinatorics
publication date: 2000-11-30T00:00:00Z
full work available at URL: https://eudml.org/doc/120810
full work available at URL: v7i1r51.html
review text: A graph \(\Gamma\) with diameter \(d\) is strongly distance-regular if \(\Gamma\) is distance-regular and its distance-\(d\) graph is strongly regular. It has been conjectured that a strongly distance-regular graph is antipodal or has diameter at most three. Let \(\Gamma\) be a graph with spectrum \(\{\lambda_0^{m_0},\dots,\lambda_d^{m_d}\}\), where \(\lambda_0>\cdots >\lambda_d\), and let \(n=|V(\Gamma)|\). Let \(\pi_i=\prod_{j\neq i}|\lambda_i-\lambda_j|\), \(\sigma_e=m_2+m_4+\cdots \), \(\sigma_o=m_1+m_3+\cdots \), \(\Sigma_e=\pi_0/\pi_2+\pi_0/\pi_4+\cdots \), \(\Sigma_o=\pi_0/\pi_1+\pi_0/\pi_3+\cdots \), for \(u\in V\) let \(k_{d-1}(u)= |\Gamma_{d-1}(u)|\) and let \(H={\displaystyle \frac{n}{\sum_{u\in V}1/k_{d-1}(u)}}\). The main result of this paper is the following Theorem 2.2: A regular graph \(\Gamma\) with \(n\) vertices, eigenvalues \(\lambda_0>\cdots >\lambda_d\), and parameters \(\sigma_e\) and \(\Sigma_e\) as above, is strongly distance-regular if and only if \[ H=n-\frac{n\sigma_e \sigma_o}{n\Sigma_e \Sigma_o+(\sigma_e-\Sigma_e) (\sigma_o+\Sigma_o)}. \]
Mathematics Subject Classification ID: 05E30
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 05C75
zbMATH DE Number: 1522360
zbMATH Keywords: distance-regular graphs
zbMATH Keywords: strongly regular graph
zbMATH Keywords: spectral characterization
reviewed by: Alexandre A. Makhnev
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1522360