On Zagreb indices of graphs

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 $\mathrm{\mid }A\mathrm{\mid }+\mathrm{\mid }B\mathrm{\mid }<\mathrm{\mid }C\mathrm{\mid }$. Finally, we give a conjecture $\mathrm{\mid }A\mathrm{\mid }<\mathrm{\mid }B\mathrm{\mid }$.