Transitive Permutation Groups with Non-self-paired Suborbits of Length 2 and Their Graphs

Transitive permutation groups having a non self paired suborbit of length are investigated via the corresponding orbital graphs If G is such a group and X is the orbital graph associated with a sub orbit of length of G which is not self paired then X has valency and admits a vertex and edge but not arc transitive action of G There is a natural balanced orientation of the edge set of X induced and preserved by G An analysis of the properties of this oriented graph is performed using a variety of graph theoretic tools resulting in some partial results on the point stabilizer of G in the case when X is connected In particular the point stabilizer must be a group generated by h involutions Moreover a characterization of the groups G is obtained in the case of point stabilizers of order and in the case of abelian point stabilizers Finally a graphical realization of such actions is given that is an in nite family of tetravalent graphs admitting a vertex and edge but not arc transitive action with vertex stabilizers Z h h as well as an in nite family of such graphs with vertex stabilizer D the dihedral group of order is constructed