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

Abstract

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 \({\frac{r}{2}K_2}\) or \({\frac{r-3}{2}K_2+K_3}\).