Finite 2-Geodesic Transitive Abelian Cayley Graphs

In this paper, we first give a classification of the family of 2-geodesic transitive abelian Cayley graphs. Let (Gamma ) be such a graph which is not 2-arc transitive. It is shown that one of the following holds: (1) (Gamma cong mathrm{K}_{m[b]}) for some (mge 3) and (bge 2); (2) (Gamma ) is a normal Cayley graph of an elementary abelian group; (3) (Gamma ) is a cover of Cayley graph (Gamma _K) of an abelian group T / K, where either (Gamma _K) is complete arc transitive or (Gamma _K) is 2-geodesic transitive of girth 3, and A / K acts primitively on (V(Gamma _K)) of type Affine or Product Action. Second, we completely determine the family of 2-geodesic transitive circulants.