Symmetric graphs with complete quotients

Abstract

Let Γ be a G-symmetric graph with vertex set V . We suppose that V admits a Ginvariant partition B = {B = B 0 , B 1 , . . . , B b }, with parts B i of size v, and that the quotient graph Γ B induced on B is a complete graph K b+1 . Then, for each pair of suffices i, j (i = j), the graph B i , B j induced on B i ∪ B j is bipartite with each vertex of valency 0 or t (a constant). When t = 1, it was shown earlier how a flag-transitive 1-design D(B) induced on the part B can sometimes be used to classify possible triples (Γ, G, B). Here we extend these ideas to t ≥ 1 and prove that, if G(B) B is 2-transitive and the blocks of D(B) have size less than v, then either (i) v < b, or (ii) the triple (Γ, G, B) is known explicitly.