A GEOMETRICAL APPROACH TO IMPRIMITIVE GRAPHS

We establish a geometrical framework for the study of imprimitive, G-s ymmetric graphs Gamma by exploiting the fact that any G-partition B of the vertex set V Gamma gives rise both to a quotient graph Gamma(B) a nd to a tactical configuration D(B) induced on each block B epsilon B. We also examine those cases in which D(B) is degenerate, and characte rize the possible graphs Gamma in many cases where the quotient Gamma( B) is either a complete graph or a circuit. When D(B) is non-degenerat e, a natural extremal case occurs when D(B) is a symmetric 2-design wi th stabilizer G(B) acting doubly transitively on points: we characteri ze such graphs in the case where Gamma(B) is complete.