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GraphTriangleCount[g]

gives the number of triangles in the graph g.

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Scope  
Applications  
Properties & Relations  
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GraphTriangleCount

GraphTriangleCount[g]

gives the number of triangles in the graph g.

Details

Examples

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Basic Examples  (2)

Count triangles in a graph:

Wolfram Language code: g = [image];
Wolfram Language code: GraphTriangleCount[g]

Count triangles in social networks:

Wolfram Language code: g = ExampleData[{"NetworkGraph", "ZacharyKarateClub"}];
Wolfram Language code: GraphTriangleCount[g]

Scope  (4)

GraphTriangleCount works with undirected graphs:

Wolfram Language code: GraphTriangleCount[[image]]

Directed graphs:

Wolfram Language code: GraphTriangleCount[[image]]

Use rules to specify the graph:

Wolfram Language code: GraphTriangleCount[{1 -> 2, 1 -> 3, 2 -> 3, 2 -> 4, 3 -> 1, 3 -> 4}]

GraphTriangleCount works with large graphs:

Wolfram Language code: g = ExampleData[{"NetworkGraph", "WorldWideWeb"}];
Wolfram Language code: EdgeCount[g]
Wolfram Language code: GraphTriangleCount[g]//Timing

Applications  (1)

Compute the transitivity of a graph:

Wolfram Language code: g = ExampleData[{"NetworkGraph", "USPoliticsBooks"}];
Wolfram Language code: 3 GraphTriangleCount[g] / Sum[d (d - 1) / 2, {d, VertexDegree[g]}]

Compare with GlobalClusteringCoefficient:

Wolfram Language code: GlobalClusteringCoefficient[g]

Properties & Relations  (2)

Use MatrixPower to count triangles in a graph:

Wolfram Language code: g = ExampleData[{"NetworkGraph", "FamilyGathering"}];
Wolfram Language code: Tr[MatrixPower[AdjacencyMatrix[g], 3]] / 6
Wolfram Language code: GraphTriangleCount[g]

Use FindCycle to list triangles in a graph:

Wolfram Language code: g = ExampleData[{"NetworkGraph", "FamilyGathering"}];
Wolfram Language code: FindCycle[g, {3}, All]//Short
Wolfram Language code: Length[%]
Wolfram Language code: GraphTriangleCount[g]

Possible Issues  (1)

Multi-edges are ignored:

Wolfram Language code: GraphTriangleCount[[image]]
Wolfram Research (2024), GraphTriangleCount, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphTriangleCount.html.

Text

Wolfram Research (2024), GraphTriangleCount, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphTriangleCount.html.

CMS

Wolfram Language. 2024. "GraphTriangleCount." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/GraphTriangleCount.html.

APA

Wolfram Language. (2024). GraphTriangleCount. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GraphTriangleCount.html

BibTeX

@misc{reference.wolfram_2026_graphtrianglecount, author="Wolfram Research", title="{GraphTriangleCount}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/GraphTriangleCount.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_graphtrianglecount, organization={Wolfram Research}, title={GraphTriangleCount}, year={2024}, url={https://reference.wolfram.com/language/ref/GraphTriangleCount.html}, note=[Accessed: 12-June-2026]}