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CompleteGraphQ[g]

yields True if the graph g is a complete graph, and False otherwise.

CompleteGraphQ[g,vlist]

yields True if the subgraph induced by vlist is a complete graph, and False otherwise.

Details
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CompleteGraphQ

CompleteGraphQ[g]

yields True if the graph g is a complete graph, and False otherwise.

CompleteGraphQ[g,vlist]

yields True if the subgraph induced by vlist is a complete graph, and False otherwise.

Details

  • A graph is complete if there is an edge between every pair of distinct vertices.
  • CompleteGraphQ works with undirected graphs, directed graphs, multigraphs, and mixed graphs.

Examples

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Basic Examples  (2)

Test whether a graph is complete:

Wolfram Language code: Graph[{12, 23, 31}]
Wolfram Language code: CompleteGraphQ[%]
Wolfram Language code: CompleteGraph[5]
Wolfram Language code: CompleteGraphQ[%]
Wolfram Language code: AdjacencyGraph[{{0, 1, 1}, {1, 0, 1}, {1, 1, 0}}]
Wolfram Language code: CompleteGraphQ[%]

Petersen graph is not a complete graph:

Wolfram Language code: PetersenGraph[5, 2]
Wolfram Language code: CompleteGraphQ[%]

Scope  (6)

Test undirected or directed graphs:

Wolfram Language code: CompleteGraphQ /@ {[image], [image]}

Multigraphs:

Wolfram Language code: CompleteGraphQ[[image]]

Mixed graphs:

Wolfram Language code: CompleteGraphQ[[image]]

Test subgraphs:

Wolfram Language code: Graph[{12, 23, 31, 34}, VertexShapeFunction -> "Name"]
Wolfram Language code: {CompleteGraphQ[%, {1, 2, 3}], CompleteGraphQ[%, {1, 3, 4}]}

CompleteGraphQ gives False for anything that is not a complete graph:

Wolfram Language code: CompleteGraphQ[x]
Wolfram Language code: CompleteGraphQ[Graph[garbage]]

Test large graphs:

Wolfram Language code: g = CompleteGraph[10000];
Wolfram Language code: CompleteGraphQ[g]//Timing
Wolfram Language code: {VertexCount[g], EdgeCount[g]}

Properties & Relations  (11)

A complete graph has no self-loops:

Wolfram Language code: CompleteGraph[4]
Wolfram Language code: {CompleteGraphQ[%], LoopFreeGraphQ[%]}

A TreeGraph is not a complete graph:

Wolfram Language code: TreeGraph[{12, 13}]
Wolfram Language code: CompleteGraphQ[%]

The only complete path graphs are triangles (undirected):

Wolfram Language code: PathGraph[{12, 23, 31}]
Wolfram Language code: CompleteGraphQ[%]

And the directed path:

Wolfram Language code: PathGraph[{12, 21}]
Wolfram Language code: CompleteGraphQ[%]

A complete graph on vertices has edges:

Wolfram Language code: EdgeCount[CompleteGraph[n]]

The complete graph is a cycle graph :

Wolfram Language code: {CompleteGraph[3], CycleGraph[3]}

The complete graph is a wheel graph :

Wolfram Language code: {CompleteGraph[4], WheelGraph[4]}

The complete graph is the line graph of the star graph :

Wolfram Language code: {CompleteGraph[6], LineGraph[StarGraph[7]]}

A complete graph is an -regular graph:

Wolfram Language code: VertexDegree[CompleteGraph[5]]

The GraphComplement of a complete graph is an empty graph:

Wolfram Language code: GraphComplement[CompleteGraph[5]]
Wolfram Language code: EmptyGraphQ[%]

For a complete graph, all entries outside the diagonal are 1s in the AdjacencyMatrix:

Wolfram Language code: AdjacencyMatrix[CompleteGraph[25]]//MatrixPlot

Complete graphs are their own cliques:

Wolfram Language code: CompleteGraph[5]
Wolfram Language code: Subgraph[%, First[FindClique[%]]]
Wolfram Research (2010), CompleteGraphQ, Wolfram Language function, https://reference.wolfram.com/language/ref/CompleteGraphQ.html (updated 2014).

Text

Wolfram Research (2010), CompleteGraphQ, Wolfram Language function, https://reference.wolfram.com/language/ref/CompleteGraphQ.html (updated 2014).

CMS

Wolfram Language. 2010. "CompleteGraphQ." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/CompleteGraphQ.html.

APA

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

BibTeX

@misc{reference.wolfram_2026_completegraphq, author="Wolfram Research", title="{CompleteGraphQ}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/CompleteGraphQ.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_completegraphq, organization={Wolfram Research}, title={CompleteGraphQ}, year={2014}, url={https://reference.wolfram.com/language/ref/CompleteGraphQ.html}, note=[Accessed: 13-June-2026]}