On graphs in which the neighborhoods of vertices are strongly regular with eigenvalue 2
type: Item
instance of: scholarly article
title: On graphs in which the neighborhoods of vertices are strongly regular with eigenvalue 2
zbMATH Open document ID: 1258.05073
DOI: 10.1134/S1064562412030179
published in: Doklady Mathematics
publication date: 2012-11-15T00:00:00Z
review text: Let \(\Gamma\) be a simple graph and \(a\) be a fixed vertex in \(\Gamma\). Let \(\Gamma_i(a)\) denote the \(i\)-neighborhood of \(a\). \(\Gamma\) is called an amply regular graph with parameters \((v,k,\lambda,\mu)\) if \(\Gamma\) is a regular graph of degree \(k\) on \(v\) vertices in which each edge lies in \(\lambda\) triangles and \(\Gamma_1(a) \cup \Gamma_1(b)\) contains \(\mu\) vertices for any two vertices \(a\) and \(b\) separated by a distance of \(2\) in \(\Gamma\). An amply regular graph of distance \(2\) is called a strongly regular graph. Further, let \(\Gamma\) be a connected amply regular graph in which the neighborhoods of vertices are strongly regular graphs with parameters \((v',k',\lambda',\mu')\) described in this work. It is demonstrated that if \(\lambda' \geq 2\) then one of the following assertions holds: {\parindent=7mm \begin{itemize} \item[(1)]\(\Gamma\) is a strongly regular graph with parameters \((100,36,14,12)\) in which the neighborhoods of vertices have the parameters \((36,14,4,6)\); \item[(2)]diameter \(d(\Gamma) = 3\) and either \begin{itemize} \item[(i)]\(\mu = 18,24,36\) and the neighborhoods of vertices have the parameters \((105,32,4,12)\); \item[(ii)]\(\mu = 35\) and the neighborhoods of vertices have the parameters \((162,56,10,24)\); \item[(iii)]\(\mu = 66\) and the neighborhoods of vertices have the parameters \((243,110,37,60)\) or \item[(iv)]\(\mu = 70\) and the neighborhoods of vertices have the parameters \((253,112,36,60)\). \end{itemize} \end{itemize}}
reviewed by: Mirko Lepović
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 05C30
zbMATH DE Number: 6106462
zbMATH Keywords: strongly regular graph, amply regular graph
zbMATH Keywords: Taylor graph
author: Alexandre A. Makhnev
author: Marina Sefovna Nirova
MaRDI profile type: Publication
cites work: Extensions of strongly regular graphs with eigenvalue 2
cites work: Partial geometries and their extensions
description: scientific article; zbMATH DE number 6106462