On graphs in which the neighborhoods of vertices are strongly regular with eigenvalue 2
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}}
strongly regular graph, amply regular graph
scientific article; zbMATH DE number 6106462