Editing Complement graph (section)

==Definition==
Let {{math|1=''G''&nbsp;=&nbsp;(''V'',&nbsp;''E'')}} be a [[simple graph]] and let {{mvar|K}} consist of all 2-element subsets of {{mvar|V}}. Then {{math|1=''H''&nbsp;=&nbsp;(''V'',&nbsp;''K''&nbsp;\&nbsp;''E'')}} is the complement of {{mvar|G}},<ref>{{Citation
| last=Diestel | first=Reinhard
| title=Graph Theory
| publisher=[[Springer Science+Business Media|Springer]]
| year=2005
| edition=3rd
| isbn=3-540-26182-6
}}. [http://diestel-graph-theory.com/index.html Electronic edition], page 4.</ref> where {{math|''K''&nbsp;\&nbsp;''E''}} is the [[relative complement]] of {{mvar|E}} in {{mvar|K}}. For [[directed graph]]s, the complement can be defined in the same way, as a directed graph on the same vertex set, using the set of all 2-element [[ordered pair]]s of {{mvar|V}} in place of the set {{mvar|K}} in the formula above. In terms of the [[adjacency matrix]] ''A'' of the graph, if ''Q'' is the adjacency matrix of the [[complete graph]] of the same number of vertices (i.e. all entries are unity except the diagonal entries which are zero), then the adjacency matrix of the complement of ''A'' is ''Q-A''.

The complement is not defined for [[multigraph]]s. In graphs that allow [[loop (graph theory)|self-loops]] (but not multiple adjacencies) the complement of {{mvar|G}} may be defined by adding a self-loop to every vertex that does not have one in {{mvar|G}}, and otherwise using the same formula as above. This operation is, however, different from the one for simple graphs, since applying it to a graph with no self-loops would result in a graph with self-loops on all vertices.