The complement of a graph 
, sometimes called the edge-complement (Gross and Yellen 2006,
p. 86), is the graph 
, sometimes denoted 
or 
(e.g., Clark and Entringer 1983), with the same vertex
set but whose edge set consists of the edges not
present in 
(i.e., the complement of the edge set of 
with respect to all possible edges on the vertex
set of 
).
The graph sum 
on a 
-node graph 
is therefore the complete graph 
, as illustrated above.
The adjacency matrix 
of the graph complement of the graph with adjacency
matrix 
is given by
 |
(Ellis-Monaghan and Merino 2008), where 
is the unit matrix and 
is the identity
matrix.
A graph complement can be computed in the Wolfram Language by the command GraphComplement[g].
