The Automorphism Groups of Certain Tetravalent Metacirculant Graphs

Abstract

Recently, in connection with the classification problem for non-Cayley tetravalent metacirculant graphs, three families of special tetravalent metacirculant graphs, denoted by \(\Phi_1, \Phi_2\), and \(\Phi_3\), have been defined [11]. It has also been shown [11] that any non-Cayley tetravalent metacirculant graph is isomorphic to a union of disjoint copies of a graph in one of the families \(\Phi_1, \Phi_2\), or \(\Phi_3\). A natural question raised from the result is whether all graphs in these families are non-Cayley. In this paper we determine the automorphism groups of all graphs in the family \(\Phi_2\). As a corollary, we show that every graph in \(\Phi_2\) is a connected non-Cayley tetravalent metacirculant graph.