Denote $A(G)=A$. Let $H$ be the subgraph for which $\lceil \frac{|E_H|}{|V_H|-1}\rceil=A$. Write $e$ for $|E_H|$, $v$ for $|V_H|$. We have $e/(v-1)>A-1$, thus $e>(v-1)(A-1)$. On the other hand, $e\leqslant v(v-1)/2$. Therefore $v/2>A-1$, $v>2A-2$, $v\geqslant 2A-1$. Then $$M(G)\geqslant \frac{2e}v=\frac{2e}{v-1}\cdot \frac{v-1}v>2(A-1)\cdot \frac{2A-2}{2A-1}>2A-3,$$
thus $\lceil M(G)\rceil \geqslant 2A-2$.