The first Zagreb index for a graph with vertex count 
and vertex degrees 
for 
, ..., 
is defined by
 |
(1)
|
The notations 
(e.g., Lin et al. 2023) and 
(e.g., Devillers and Balaban 2000) are variously used.
For the cycle graph 
with 
and the complete graph 
with 
, respectively, the vertex
degrees are 2 and 
, giving
 |  |  |
(2)
|
 |  |  |
(3)
|
The first Zagreb index is related to earlier indices 
(defined by Platt) and 
(defined by Gordon and Scantlebury) by
 |
(4)
|
where 
is the edge count of a graph 
(Devillers and Balaban 2000, p. 27).
