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SimpleGraph[g]

gives the underlying simple graph from the graph g.

SimpleGraph[{vw,…}]

uses rules vw to specify the graph g.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Properties & Relations  
See Also
Related Guides
History
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SimpleGraph

SimpleGraph[g]

gives the underlying simple graph from the graph g.

SimpleGraph[{vw,…}]

uses rules vw to specify the graph g.

Details and Options

  • SimpleGraph is also known as strict graph.
  • SimpleGraph[g] removes all self-loops and multiple edges between the same vertices.
  • SimpleGraph preserves directed edges as directed. UndirectedGraph can be used to compute the underlying undirected graph.
  • SimpleGraph works with undirected graphs, directed graphs, multigraphs, and mixed graphs.

Examples

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Basic Examples  (1)

Remove self-loops from a graph:

Wolfram Language code: SimpleGraph[[image]]

Scope  (6)

SimpleGraph works with undirected graphs:

Wolfram Language code: SimpleGraph[[image]]

Directed graphs:

Wolfram Language code: SimpleGraph[[image]]

Multigraph:

Wolfram Language code: SimpleGraph[[image]]

Mixed graph:

Wolfram Language code: SimpleGraph[[image]]

Use rules to specify the graph:

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

SimpleGraph works with large graphs:

Wolfram Language code: g = AdjacencyGraph[SparseArray[{{i_, i_} -> 1, {i_, j_} /; j == i + 1 -> 1}, {10 ^ 6, 10 ^ 6}]];
Wolfram Language code: Timing[h = SimpleGraph[g];]
Wolfram Language code: SimpleGraphQ /@ {g, h}
Wolfram Language code: EdgeCount /@ {g, h}

Properties & Relations  (7)

A graph with self-loops is not simple:

Wolfram Language code: Graph[{12, 23, 31, 11}]
Wolfram Language code: {SimpleGraphQ[%], LoopFreeGraphQ[%]}

A PathGraph is always simple:

Wolfram Language code: g = PathGraph[Range[20]]
Wolfram Language code: {PathGraphQ[g], SimpleGraphQ [g]}

A TreeGraph without multiple edges is simple:

Wolfram Language code: g = CompleteKaryTree[3, 3]
Wolfram Language code: {TreeGraphQ[g], SimpleGraphQ [g]}

The adjacency matrix of a simple graph has entries not greater than 1:

Wolfram Language code: g = CycleGraph[4]
Wolfram Language code: SimpleGraphQ[g]
Wolfram Language code: (m = AdjacencyMatrix[g])//MatrixForm

Diagonal elements are all zeros:

Wolfram Language code: Diagonal[m]//Normal

The incidence matrix of a simple graph has entries -1, 0, or 1 and no repeated column:

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

All vertices of a simple graph have a maximum degree less than the number of vertices:

Wolfram Language code: g = WheelGraph[6]
Wolfram Language code: SimpleGraphQ[g]
Wolfram Language code: Max[VertexDegree[g]] < VertexCount[g]

A nontrivial simple graph must have at least one pair of vertices with the same degree:

Wolfram Language code: g = GridGraph[{2, 3}]
Wolfram Language code: SimpleGraphQ[g]
Wolfram Language code: VertexDegree[g]
Wolfram Research (2010), SimpleGraph, Wolfram Language function, https://reference.wolfram.com/language/ref/SimpleGraph.html (updated 2015).

Text

Wolfram Research (2010), SimpleGraph, Wolfram Language function, https://reference.wolfram.com/language/ref/SimpleGraph.html (updated 2015).

CMS

Wolfram Language. 2010. "SimpleGraph." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/SimpleGraph.html.

APA

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

BibTeX

@misc{reference.wolfram_2026_simplegraph, author="Wolfram Research", title="{SimpleGraph}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/SimpleGraph.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_simplegraph, organization={Wolfram Research}, title={SimpleGraph}, year={2015}, url={https://reference.wolfram.com/language/ref/SimpleGraph.html}, note=[Accessed: 13-June-2026]}