TOPOLOGICAL COVERS OF COMPLETE GRAPHS

Let Gamma be a connected G-symmetric graph of valency r, whose vertex set V admits a non-trivial G-partition B, with blocks B epsilon B of s ize v and with k less than or equal to v independent edges joining eac h pair of adjacent blocks. In a previous paper we introduced a framewo rk for analysing such graphs Gamma in terms of (a) the natural quotien t graph Gamma(B) of valency b = vr/k, and (b) the 1-design D(B) induce d on each block. Here we examine the case where k = v and Gamma(B) = K b+1 is a complete graph. The 1-design D(B) is then degenerate, so give s no information: we therefore make the additional assumption that the stabilizer G(B) of the block B acts 2-transitively on B. We prove tha t there is then a unique exceptional graph for which \B\ = v > b + 1.