Summary: 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: \[A=\left\{G\in\mathcal{G}_n:\frac{M_1(G)}{n}>\frac{M_2(G)}{m} \right\},\ B=\left\{G\in\mathcal{G}_n:\frac{M_1(G)}{n}= \frac{M_2(G)}{m}\right\}\] and \[C=\left\{G\in\mathcal{G}_n: \frac{M_1(G)}{n}<\frac{M_2(G)}{m}\right\}.\] In this paper we prove that \(|A|+|B|<|C|\). Finally, we give a conjecture \(|A|<|B|\).