Antipodal covering graphs

A graph $\text{Γ}$ of valency k with a group G of automorphisms may be studied via the action of G on the vertex set $\text{V}\text{Γ}$. If G acts transitively on $\text{V}\text{Γ}$, then the notions of primitivity and imprimitivity are meaningful. We consider a natural notion of “block system” for a general graph $\text{Γ}$, which allows us to derive a “quotient” graph Γ whose vertices correspond to the blocks. The ideas are applied to antipodal systems in antipodal graphs: in particular we prove that for an antipodal distance-regular graph, the block size r cannot exceed the valency k; we further show that an antipodal distance-regular graph with r = k is (i) a circuit graph, (ii) a complete bipartite graph, or (iii) a threefold covering of Tutte's trivalent eight-cage.