Automorphism groups of tetravalent Cayley graphs on minimal non-abelian groups

Abstract

A Cayley graph \(X\)\(=\)Cay\((G,S)\) is called {\it normal} for \(G\) if the right regular representation \(R(G)\)  of \(G\) is normal in the full automorphism group Aut\((X)\) of \(X\). In the present paper it is proved that all connected tetravalent Cayley graphs on a minimal non-abelian group \(G\) are normal when \((|G|, 2)=(|G|,3)=1\), and \(X\) is not isomorphic to either Cay\((G,S)\), where \(|G|=5^n\), and \(|\)Aut(X)\(|\)\(=\)\(2^m.3.5^n\), where \(m \in \{2,3\}\) and \(n\geq 3\), or Cay\((G,S)\) where \(|G|=5q^n\) (\(q\) is prime) and \(|{\hbox{Aut}}(X)|=2^m.3.5.q^n\), where \(q\geq 7\),  \(m \in \{2,3\}\) and \(n\geq 1\).