# GraphComplement

gives the graph complement of the graph g.

GraphComplement[{vw,}]

uses rules vw to specify the graph g.

# Details and Options

• GraphComplement is also known as edge-complementary graph.
• The graph complement has the same vertices and edges defined by two vertices being adjacent only if they are not adjacent in g.
• GraphComplement works with undirected graphs, directed graphs, multigraphs, and mixed graphs.

# Examples

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## Basic Examples(2)

Graph complement of cycle graphs:

Graph complement of directed graphs:

## Scope(6)

GraphComplement works with undirected graphs:

Directed graphs:

Multiple graphs:

Mixed graphs:

Use rules to specify the graph:

GraphComplement works with large graphs:

## Properties & Relations(7)

The complement of a CompleteGraph is an edgeless graph:

The complement of the complement is the original graph (for simple graphs):

The complement of the graph can be obtained from its adjacency matrix:

An independent vertex set of the graph is a clique of its complement graph:

The complement of the line graph of is a Petersen graph:

The graph union of any simple graph and its complement is a complete graph:

The graph intersection of any graph and its complement is an empty graph:

Wolfram Research (2010), GraphComplement, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphComplement.html (updated 2015).

#### Text

Wolfram Research (2010), GraphComplement, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphComplement.html (updated 2015).

#### CMS

Wolfram Language. 2010. "GraphComplement." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/GraphComplement.html.

#### APA

Wolfram Language. (2010). GraphComplement. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GraphComplement.html

#### BibTeX

@misc{reference.wolfram_2024_graphcomplement, author="Wolfram Research", title="{GraphComplement}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/GraphComplement.html}", note=[Accessed: 17-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_graphcomplement, organization={Wolfram Research}, title={GraphComplement}, year={2015}, url={https://reference.wolfram.com/language/ref/GraphComplement.html}, note=[Accessed: 17-September-2024 ]}