Comparing Zagreb indices and variable Zagreb indices
Damir Vukicevic – University of Split, Croatia
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.