Two-geodesic-transitive graphs which are locally connected

We investigate the family of 2-geodesic-transitive graphs which are locally connected. Let Γ be such a graph. It is first shown that: for any integer d≥2, there exists such a Γ of diameter d; for any integer k≥3, there exists such a Γ of valency k unless k is a prime and k≡3(mod4). Next, we completely determine the family of 2-geodesic-transitive graphs which are locally isomorphic to mCn¯ for some m≥1,n≥3. Finally, we give a reduction result for the family of locally connected (G,2)-geodesic-transitive graphs.