On automorphisms of a strongly regular graph with parameters \((96, 45, 24, 18)\)
Let \(\Gamma\) be a strongly regular graph with parameters \((96,45,24,18)\), \(g\) be an element of prime order \(p\) from \(\mathrm{Aut}(\Gamma)\), and \(\Delta = \mathrm{Fix}(g)\). Then one of the following assertion holds: {\parindent=6mm \begin{itemize} \item[(1)]\(\Delta\) is an empty graph, and either \(p=2\) and \(\alpha_1(g)\in \{0,36,72\}\) or \(p=2\) and \(\alpha_1(g) \in \{0,24,48,72,96\}\). \item[(2)]\(\Delta\) is a \(\beta\)-clique and either \begin{itemize} \item[(i)]\(p=2\), \(\beta\) is even, \(4\leq \beta\leq 16\) and \(3\beta+\alpha_1(g)\) is divided by 24 or \item[(ii)]\(p=5\); \(\beta=1\) and \(\alpha_1(g)=45\), \(\beta=6\) and there are 18 clique \(\langle g\rangle\)-orbits on \(\Gamma-\Delta\), \(\beta=11\) and there are 13 clique and 4 pentagonal \(\langle g\rangle\)-orbits on \(\Gamma-\Delta\), or \(\beta=16\) and there are eight clique and eight pentagonal \(\langle g\rangle\)-orbits on \(\Gamma-\Delta\). \end{itemize} \item[(3)]\(\Delta\) is a \(\gamma\)-coclique, \(p=3\), and either \(\gamma=3\) and \(\alpha_1(g)\in \{27,63\}\) or \(\gamma=6\) and \(\alpha_1(g)\in \{18, 54, 90\}\). \item[(4)]\(p=3\), and either \(|\Delta|\in\{6,9,12\}\) and \(3|\Delta|+\alpha_1(g)\) is divided by 36 or \(|\Delta|=24\) and \(\alpha_1(g)=72\). \item[(5)]\(p=2\), and either \(4\leq |\Delta|\leq 40\) and \(3|\Delta|+\alpha_1(g)\) is divided by 24 or \(|\Delta|=48\), any vertex in \(\Delta\) is adjacent to a vertex in \(\Gamma-\Delta\), and \(\alpha_1(g)=48\). \end{itemize}}
scientific article; zbMATH DE number 6124961