Possible numbers of ones in the squares of 0–1 matrices with a given rank

For a symmetric 0–1 matrix A, we give the number of ones in A 2 when rank(A) = 1, 2, and give the maximal number of ones in A 2 when rank(A) = k (3 ≤ k ≤ n). The sufficient and necessary condition under which the maximal number is achieved is also obtained. For generic 0–1 matrices, we only study the cases of rank 1 and rank 2.