Groupe d’études et de recherche en analyse des décisions

# Comparing Zagreb indices and variable Zagreb indices

## Damir Vukicevic – University of Split, Croatie

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.