Graphs with given group (Q1813925)

Let a finite group \(G\) and an integer \(n\geq 3\) be given. \textit{G. Sabidussi} [Can. J. Math. 9, 515--525 (1957; Zbl 0079.39202)] proved the existence of infinitely many connected \(n\)-valent graphs \(\Gamma\) with \(\Aut(\Gamma)\cong G\). The present paper extends Sabidussi's result: given any positive integer \(t\), if \(n\geq\max\{| G|,3\}\), then the same conclusion may be drawn with the further properties that \(\Aut(\Gamma)\) fixes pointwise some \(t\)-subset of \(V(\Gamma)\) and acts semi-regularly on \(V(\Gamma)-T\).