  Let ${\mathcal G}_n$ be the set of class of graphs of order $n$. The first
Zagreb index $M_1(G)$ is equal to the sum of squares of the degrees of the
vertices, and the second Zagreb index $M_2(G)$ is equal to the sum of the
products of the degrees of pairs of adjacent vertices of the underlying
molecular graph $G$. The three set of graphs are as follows: \begin{eqnarray*}
&&A=\left\{G\in {\mathcal
G}_n:\,\frac{M_1(G)}{n}>\frac{M_2(G)}{m}\right\},~B=\left\{G\in {\mathcal
G}_n:\,\frac{M_1(G)}{n}=\frac{M_2(G)}{m}\right\} \mbox{ and }&&
&&~~~~~~~~~~~~~~~~~~~~~~~~~C=\left\{G\in {\mathcal
G}_n:\,\frac{M_1(G)}{n}<\frac{M_2(G)}{m}\right\}. \end{eqnarray*} In this paper
we prove that $|A|+|B|<|C|$. Finally, we give a conjecture $|A|<|B|$.
