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LoopFreeGraphQ[g]

yields True if the graph g has no self-loops, and False otherwise.

Details
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Examples  
Basic Examples  
Scope  
Properties & Relations  
Possible Issues  
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LoopFreeGraphQ

LoopFreeGraphQ[g]

yields True if the graph g has no self-loops, and False otherwise.

Details

  • A self-loop in a graph is an edge that connects a vertex to itself.

Examples

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Basic Examples  (2)

Test whether a graph has no self-loops:

Wolfram Language code: GraphData["CubicalGraph"]
Wolfram Language code: LoopFreeGraphQ[%]

LoopFreeGraphQ gives False for a graph with self-loops:

Wolfram Language code: LoopFreeGraphQ[[image]]

Scope  (6)

LoopFreeGraphQ works with undirected graphs:

Wolfram Language code: LoopFreeGraphQ[[image]]

Directed graphs:

Wolfram Language code: LoopFreeGraphQ[[image]]

Multigraphs:

Wolfram Language code: LoopFreeGraphQ[[image]]

Mixed graphs:

Wolfram Language code: LoopFreeGraphQ[[image]]

LoopFreeGraphQ gives False for anything that is not a graph without self-loops:

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

LoopFreeGraphQ works with large graphs:

Wolfram Language code: GridGraph[{10, 10, 10, 10}];
Wolfram Language code: LoopFreeGraphQ[%]//Timing

Properties & Relations  (5)

A bipartite graph has no self-loops:

Wolfram Language code: Graph[{12, 11}]
Wolfram Language code: {BipartiteGraphQ[%], LoopFreeGraphQ[%]}

A TreeGraph has no self-loops:

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

A typical PathGraph has no self-loops:

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

The adjacency matrix of a graph without self-loops has a zero diagonal:

Wolfram Language code: g = PetersenGraph[5, 2]
Wolfram Language code: LoopFreeGraphQ[g]
Wolfram Language code: Diagonal[AdjacencyMatrix[g]]//Normal

The incidence matrix of a graph without self-loops has no entries greater than 1:

Wolfram Language code: g = Graph[{12, 23, 31}]
Wolfram Language code: LoopFreeGraphQ[g]
Wolfram Language code: IncidenceMatrix[g]//MatrixForm

Possible Issues  (1)

LoopFreeGraphQ gives False for non-explicit graphs:

Wolfram Language code: LoopFreeGraphQ[CompleteGraph[n]]
Wolfram Research (2010), LoopFreeGraphQ, Wolfram Language function, https://reference.wolfram.com/language/ref/LoopFreeGraphQ.html.

Text

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

CMS

Wolfram Language. 2010. "LoopFreeGraphQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/LoopFreeGraphQ.html.

APA

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

BibTeX

@misc{reference.wolfram_2026_loopfreegraphq, author="Wolfram Research", title="{LoopFreeGraphQ}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/LoopFreeGraphQ.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_loopfreegraphq, organization={Wolfram Research}, title={LoopFreeGraphQ}, year={2010}, url={https://reference.wolfram.com/language/ref/LoopFreeGraphQ.html}, note=[Accessed: 13-June-2026]}