Constructing a Class of Symmetric Graphs

We find a natural construction of a large class of symmetric graphs from pointand block-transitive 1-designs. The graphs in this class can be characterized as G-symmetric graphs whose vertex sets admit a G-invariant partition B of block size at least 3 such that, for any two blocks B,C of B, either there is no edge between B and C , or there exists only one vertex in B not adjacent to any vertex in C . The special case where the quotient graph 0B of 0 relative to B is a complete graph occurs if and only if the 1-design needed in the construction is a G-doubly transitive and G-block-transitive 2design, and in this case we give an explicit classification of 0 when G is a doubly transitive projective group or an affine group containing the affine general group. Examples of such graphs include cross ratio graphs studied recently by Gardiner, Praeger and Zhou and some other graphs with vertices the (point, line)-flags of the projective or affine geometry.