# FindKClique

FindKClique[g,k]

finds a largest k-clique in the graph g.

FindKClique[g,k,n]

finds a k-clique containing at most n vertices.

FindKClique[g,k,{n}]

finds a k-clique containing exactly n vertices.

FindKClique[g,k,{nmin,nmax}]

finds a k-clique containing between nmin and nmax vertices.

FindKClique[g,k,nspec,s]

finds at most s k-cliques.

FindKClique[{g,v},k,]

finds k-cliques that include the vertex v only.

FindKClique[{vw,},]

uses rules vw to specify the graph g.

# Details

• A k-clique is a maximal set of vertices that are at a distance no greater than k from each other.
• FindKClique returns a list of k-cliques.
• FindKClique will return an empty list if there is no k-clique.
• FindKClique[,k,nspec,All] finds all the k-cliques.
• FindKClique works with undirected graphs, directed graphs, multigraphs, and mixed graphs.

# Background & Context

• FindKClique finds one or more k-cliques in a graph, returning them as a list of vertices. Here, a k-clique is a maximal set of vertices that are at a distance no greater than k from each other. k-cliques are used in project selection, pattern matching, finance, and network analysis.
• FindKClique can be used to find k-cliques of different sizes, from 1 to the largest possible size (in general n for a graph on n vertices). FindKClique can be used to find a single k-clique of specified size, a specified number of cliques, or all.
• 1-cliques are cliques. All k-clans are k-cliques, but the converse is not always true. Related functions include FindClique, FindKClan, FindKClub, and FindKPlex.

# Examples

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## Basic Examples(2)

Find a largest 2-clique in a graph:

Show the 2-clique:

Find all 4-cliques:

## Scope(14)

### Specification(8)

FindKClique works with undirected graphs:

Directed graphs:

Multigraphs:

Mixed graphs:

Find a largest 2-clique:

Find k-cliques for arbitrary k:

Use rules to specify the graph:

FindKClique works with large graphs:

### Enumeration(6)

A 2-clique containing exactly 4 vertices:

A 2-clique containing at most 4 vertices:

A 2-clique containing between 3 and 5 vertices:

A largest 2-clique that includes a given vertex:

Find all 2-cliques in a graph:

FindKClique gives an empty list if there is no k-clique:

## Applications(4)

Highlight all 2-cliques of size 5:

A friendship network between members of a karate club. Find the size of a largest group of people who are friends or a friend of a friend:

The largest such groups:

A network of books linked by the same buyers on Amazon.com. Find a largest selection of books including The Clinton Wars that was frequently bought by buyers who previously bought a common book:

To prevent data packets from circulating indefinitely in a mobile ad hoc network, a time to live (TTLthe maximum number of edges traversed) value is set to 3. Find all devices that can be reached from device 1:

Show the subnetwork:

The best time to live for data packets:

Unreachable devices:

## Properties & Relations(8)

A k-clique in a graph g is a clique in the graph k power of g:

A 1-clique is a clique:

A complete graph is a maximum k-clique:

A star graph is a maximum 2-clique:

The (k-1)-clique is contained in a k-clique:

All k-clans are k-cliques. The converse is not always true:

A k-club is contained in a k-clique:

The converse is not always true:

Find a largest 2-clique that includes a given vertex:

Compare with 2-clan, 2-club, and 2-plex:

Wolfram Research (2012), FindKClique, Wolfram Language function, https://reference.wolfram.com/language/ref/FindKClique.html (updated 2015).

#### Text

Wolfram Research (2012), FindKClique, Wolfram Language function, https://reference.wolfram.com/language/ref/FindKClique.html (updated 2015).

#### CMS

Wolfram Language. 2012. "FindKClique." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/FindKClique.html.

#### APA

Wolfram Language. (2012). FindKClique. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FindKClique.html

#### BibTeX

@misc{reference.wolfram_2024_findkclique, author="Wolfram Research", title="{FindKClique}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/FindKClique.html}", note=[Accessed: 16-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_findkclique, organization={Wolfram Research}, title={FindKClique}, year={2015}, url={https://reference.wolfram.com/language/ref/FindKClique.html}, note=[Accessed: 16-July-2024 ]}