Two-geodesic-transitive graphs of odd order

We study (G, 2)-geodesic-transitive graphs of odd order. We first give a reduction result on this family of graphs: Let N be an intransitive normal subgroup of G. Suppose that such a graph F is neither (G, 2)-arc-transitive nor K-m[b] where mb is odd and m, b = 3. Then, we show that F is a cover of G(N), G/N is faithful and quasiprimitive on V (G(N)), G???????(N) is (G/N, s')-geodesic-transitive of odd order and girth 3 where s' = min{2, diam(G???????(N))}. We next investigate odd order (G, 2)-geodesic-transitive graphs where G acts quasiprimitively on the vertex set and determine all the possible quasiprimitive action types and give examples for them, and we also classify the family of (G, 2)-geodesic-transitive graphs of odd order where G is primitive of product action type on the vertex set. Finally, we find all the odd order 3-geodesic-transitive graphs which are covers of complete graphs.