Since $2 |E_H|\le |V_H|(|V_H|-1)$, we get $|E_H|/(|V_H|-1)\le |E_H|/|V_H|+1/2$, taking ceiling and maximum over $H$ in LHS we get $$A(G)\le \lceil M(G)/2+1/2\rceil\le \lceil M(G)\rceil/2+1.$$
Fedor Petrov
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