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We look for a non-zero $(0, 1)$-vector in the row space of the adjacencymatrix $A(\Gamma)$ of a graph $\Gamma,$ provided $\Gamma$ has at least oneedge. Akbari, Cameron, and Khosrovshahi conjectured that there exists anon-zero $(0,1)$-vector in the row space of $A(\Gamma)$ (over the real numbers)which does not occur as a row of $A(\Gamma).$ This conjecture can be easilyverified for graphs having diameter is equal to $1$ (complete graphs). In thisarticle, we affirmatively prove this conjecture for any graph whose diameter is$\geq 4.$ Furthermore, in the remaining two cases that is, for graphs withdiameter is equal to $2$ or $3,$ we report some progress in support of theconjecture.

A conjecture of Akbari, Cameron, and Khosrovshahi on the existence of a $(0,1)$-vector in the row space of every adjacency matrix