Constructing a class of symmetric graphs

Abstract

We find a natural construction of a large class of symmetric graphs from point- and 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 ΓB of Γ 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 2-design, and in this case we give an explicit classification of Γ when G is a doubly transitive pr..