In this talk we introduce a family of finite edge-transitive graphs known as the locally s- distance transitive graphs that contains the distance transitive graphs and other important families such as locally s-arc transitive graphs of diameter at least s. We classify the graphs Γ whose subdivision graphs S(Γ) are locally s-distance transitive. We also prove a decisive relationship between the levels of arc transitivity of Γ and S(Γ). Moreover, for s ≥ 4, we show that the study of locally s-distance transitive graphs with a star quotient is equivalent to the study of a particular family of designs with strong symmetry properties.
