The maximum order of adjacency matrices of graphs with a given rank

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 conjecture and characterize G in the case that G contains an induced subgraph or .