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) \(\Gamma \cong \mathrm{K}_{mb]}\) for some \(m\ge 3\) and \(b\ge 2\); (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.