Graphs for Orthogonal Arrays and Projective Planes of Even Order

We consider orthogonal arrays of strength two and even order q having n columns which are equivalent to n − 2 mutually orthogonal Latin squares of order q. We show that such structures induce graphs on n vertices, invariant up to complementation. Previous methods worked only for single Latin squares of even order and were harder to apply. If q is divisible by four the invariant graph is simple undirected. If q is two modulo four the graph is a tournament. When n = q + 1 is maximal, the array corresponds to an affine plane, and the vertex valencies of the graph have parity q/2 modulo two. We give the graphs at all possible points and lines for twenty-two planes of order sixteen. Four of the planes, none of them of translation or dual translation type, produce non-empty graphs at some points.