Finite planes and graphs

AbstractIn an earlier paper [3], we associated to every projective plane X of order n a certain n3-dimensional vector space H(X) over GF(2) and a distinguished subset T of “triangles” in H(X); moreover, the collineation group of X is isomorphic to the subgroup of GL(H(X)) consisting of all nonsingular linear transformations ϕ with ϕ(T) = T. Here we show that T may be construed as a graph whose automorphism group is isomorphic to the collineation group of X. The graph has extra structure (“graded”, “additive”) and it can be described axiomatically, without assuming prior existence of X. We then prove that the existence of X is equivalent to the existence of the graph. Finally, we show how to imbed such an abstract graph into an abstract n3-dimensional vector space over GF(2)