Group for Research in Decision Analysis

# Comparing the Zagreb Indices

## Pierre Hansen and Damir Vukicevic

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

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

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.

, 12 pages