Graphs in which the neighborhoods of vertices are pseudogeometric graphs for \(GQ(3, 5)\)
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}}
pseudogeometric graph, generalized quadrangle
scientific article; zbMATH DE number 5998415