We look for the maximum order m(r) of the adjacency matrix A of a graph G with a fixed rank r, provided A has no repeated rows or all-zero row. Akbari, Cameron and Khosrovshahi conjecture that m(r) = 2(r 2)/2−2 if r is even, and m(r) = 5 • 2(r−3)/2 − 2 if r is odd. We prove the...