# FindKClan

FindKClan[g,k]

finds a largest k-clan in the graph g.

FindKClan[g,k,n]

finds a k-clan containing at most n vertices.

FindKClan[g,k,{n}]

finds a k-clan containing exactly n vertices.

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

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

FindKClan[g,k,nspec,s]

finds at most s k-clans.

FindKClan[{g,v},k,]

finds k-clans that include the vertex v only.

FindKClan[{vw,},]

uses rules vw to specify the graph g.

# Details

• A k-clan is a k-clique where the diameter of the corresponding subgraph is at most k.
• FindKClan returns a list of k-clans.
• FindKClan will return an empty list if there is no k-clan.
• FindKClan[,k,nspec,All] finds all the k-clans.
• FindKClan works with undirected graphs, directed graphs, multigraphs, and mixed graphs.

# Background & Context

• FindKClan finds one or more k-clans in a graph, returning them as a list of vertices. Here, a k-clan is a k-clique such that the graph diameter (i.e. the length of the longest possible path between two vertices) of the subgraph induced by the vertices is at most k. k-clans are used in project selection, pattern matching, finance, and network analysis.
• In contrast, FindKClique can be used to find k-cliques of different subgraph diameter, 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 subgraph diameter, a specified number of cliques, or all.
• 1-clans are cliques. All k-clans are k-clubs and k-cliques, but the converse is not always true. Related functions include FindClique, FindKClique, FindKClub, and FindKPlex.

# Examples

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## Basic Examples(2)

Find a largest 2-clan in a graph:

Show the 2-clan:

Find all 4-clans:

## Scope(14)

### Specification(8)

FindKClan works with undirected graphs:

Directed graphs:

Multigraphs:

Mixed graphs:

Find a largest 2-clan:

Find k-clans for arbitrary k:

Use rules to specify the graph:

FindKClan works with large graphs:

### Enumeration(6)

A 2-clan containing exactly 4 vertices:

A 2-clan containing at most 4 vertices:

A 2-clan containing between 3 and 5 vertices:

A largest 2-clan that includes a given vertex:

Find all 2-clans in a graph:

FindKClan gives an empty list if there is no clan:

## Applications(1)

Highlight all 2-clans of size 5:

## Properties & Relations(8)

Compute k-clans using FindKClique:

All 2-clans:

A 1-clan is a clique:

A complete graph is a maximum k-clan:

A star graph is a maximum 2-clan:

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

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

All k-clans are k-clubs:

The converse is not always true:

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

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

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_findkclan, author="Wolfram Research", title="{FindKClan}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/FindKClan.html}", note=[Accessed: 16-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_findkclan, organization={Wolfram Research}, title={FindKClan}, year={2015}, url={https://reference.wolfram.com/language/ref/FindKClan.html}, note=[Accessed: 16-July-2024 ]}