On finite groups with prescribed two-generator subgroups and integral Cayley graphs (Q1983955)

This paper deals with the class of finite groups in a graph theoretic context. Let \(G\) be a finite group with identity \(1\). Let us consider \(X \subset G\) be a subset such that \(1\notin X\) and \(x^{-1} \in X\) for every \(x\in X\). Then we can consider the Cayley graph \(\Gamma(G,X)\). The vertex set of such Cayley graph \(\Gamma\) is \(G\) and edges are \((g, gx)\) for \(g\in G\) and \(x \in X\). If a Cayley graph has an integral spectrum then the graph is said to be Cayley integral. Integral Cayley graphs were introduced by \textit{A. Abdollahi} and \textit{E. Vatandoost} [Electron. J. Comb. 16, No. 1, Research Paper R122, 17 p. (2009; Zbl 1186.05064)]. Cayley integral groups have been classified and it appears to be one of the following forms: \(D_6\), \(\mathrm{Dic}(\mathbb{Z}_6)\), \(\mathbb{Z}_2^m\times \mathbb{Z}_3^n\), \(\mathbb{Z}_2^m\times \mathbb{Z}_4^n\), \(\mathbb{Z}_2 \times Q_8\) \((m,n \ge 0)\). In this paper, the authors characterize the finite groups of even order with the property that for any involution \(x\) and an element \(y\) of \(G\), \(\langle x,y\rangle\) is isomorphic to one of the following groups: \(\mathbb{Z}_2\), \(\mathbb{Z}_2^2\), \(\mathbb{Z}_4\), \(\mathbb{Z}_6\), \(\mathbb{Z}_2\times \mathbb{Z}_4\), \(\mathbb{Z}_2\times \mathbb{Z}_6\) and \(A_4\). Finally, they provide a characterization for the finite groups whose Cayley graphs of degree 3 have an integral spectrum.