Comparing Zagreb indices and variable Zagreb indices

Let \(G=(V,E)\) be a simple graph with \(n=|V|\) vertices and \(m=|E|\) edges; let \(d_1, d_2, \dots, d_n\)denote the degrees of the vertices of \(G\). If \(\Delta=\max\limits_i d_i \leq 4\), \(G\) is a chemical graph. The first and second Zagreb indices are defined as

$$ M_1=\sum\limits_{i\in V} d^2_i \quad\text{ and }\quad M_2=\sum\limits_{(i,j)\in E} d_i d_j.$$

We show that for all chemical graphs \(M_1/n \leq M_2/m\). This does not hold for all general graphs, connected or not.