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Motivated by this we introduce a term $gap(G)$ defined as $gap(G)=a'(G)-\\Delta(G)$. Alon's conjecture can be rephrased as $gap(G)\\le2$ for all graphs $G$. In \\cite{manusccartprod} it was shown that $a'(G\\Box H)\\le a'(G)+a'(H)$, under some assumptions. 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