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^(th) 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:

Introduced in 2012
 (9.0)
 |
Updated in 2014
 (10.0)
2015
 (10.3)