Graphs in which the neighborhoods of vertices are pseudogeometric graphs for \(GQ(3, 5)\)
type: Item
instance of: scholarly article
title: Graphs in which the neighborhoods of vertices are pseudogeometric graphs for \(GQ(3, 5)\)
zbMATH Open document ID: 1239.05188
DOI: 10.1134/S1064562411030367
author: A. K. Gutnova
author: Alexandre A. Makhnev
published in: Doklady Mathematics
publication date: 2012-01-17T00:00:00Z
review text: A graph \(\Gamma\) is an amply regular graph with parameters \((\nu, k, \lambda, \mu)\) if it is a \(k\)-regular graph on \(\nu\) vertices such that each edge lies in \(\lambda\) triangles and any pair of vertices at distance 2 apart possess exactly \(\mu\) common neighbors. Let \(\Gamma_i(a)\) denote the set of vertices that are at distance \(i\) from the vertex \(a\). The following classification theorem is proved in this paper. Let \(\Gamma\) be a connected amply regular graph in which the neighborhoods of vertices are pseudogeometric graphs for the generalized quadrangle \(GQ(3,5)\). Then one of the following properties holds. {\parindent=6mm \begin{itemize} \item[(1)]The diameter of \(\Gamma\) is 2; \(\Gamma\) has the parameters \((245, 64, 18, 16)\) and the eigenvalues 8 and \(-6\) of multiplicities 100 and 144. \item[(2)]The diameter of \(\Gamma\) is 3; \(k_i=|\Gamma_i(a)|\) for some vertex \(a\), and either \begin{itemize} \item[(i)]\(\mu=20, k_2=144, k_3 \leq 4\), or \item[(ii)]\(\mu=18, k_2=160, 2 \leq k_3 \leq 5\). \end{itemize} \end{itemize}}
Mathematics Subject Classification ID: 05C75
Mathematics Subject Classification ID: 51E14
zbMATH DE Number: 5998415
zbMATH Keywords: strongly regular graph
zbMATH Keywords: amply regular graph
zbMATH Keywords: pseudogeometric graph, generalized quadrangle
MaRDI profile type: Publication
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cites work: The pseudo-geometric graphs for generalized quadrangles of order \((3,t)\)
cites work: Graphs in which the neighborhoods of vertices are pseudogeometric graphs for \(GQ(3,3)\)
cites work: The Gewirtz graph: An exercise in the theory of graph spectra
description: scientific article; zbMATH DE number 5998415