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 $|A|<|B|$.
Categories: math.CO