A Geometrical Approach to Imprimitive Graphs

We establish a geometrical framework for the study of imprimitive, G-symmetric graphs F by exploiting the fact that any G-partition B of the vertex set VT gives rise both to a quotient graph fB and to a tactical configuration D(B) induced on each block B e B . We also examine those cases in which D(B) is degenerate, and characterize the possible graphs f in many cases where the quotient FB is either a complete graph or a circuit. When D(fl) is non-degenerate, a natural extremal case occurs when D(B) is a symmetric 2-design with stabilizer G(B) acting doubly transitively on points: we characterize such graphs in the case where TB is complete.