Let Γ be a connected G-symmetric graph of valency r, whose vertex set V admits a non-trivial G-partition [Bscr ], with blocks B∈[Bscr ] of size v and with k[les ]v independent edges joining each pair of adjacent blocks. In a previous paper we introduced a framework for analysing such graphs Γ in terms of (a) the natural quotient graph Γ[Bscr ] of valency b=vr/k, and (b) the 1-design [Dscr ](B) induced on each block. Here we examine the case where k=v and Γ[Bscr ]=Kb+1 is a complete graph. The 1-design [Dscr ](B) is then degenerate, so gives no information: we therefore make the additional assumption that the stabilizer G(B) of the block B acts 2-transitively on B. We prove that there is then a unique exceptional graph for which [mid ]B[mid ]=v>b+1.