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GraphProduct[g1,g2]

gives the Cartesian product of the graphs g1 and g2.

GraphProduct[g1,g2,"op"]

gives the product of type "op" for the graphs g1 and g2

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Directed Graphs  
Undirected Graphs  
Mixed Graphs  
Multigraphs  
Weighted Graphs  
Special Graphs  
Properties & Relations  
See Also
Related Guides
History
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GraphProduct

GraphProduct[g1,g2]

gives the Cartesian product of the graphs g1 and g2.

GraphProduct[g1,g2,"op"]

gives the product of type "op" for the graphs g1 and g2

Details and Options

  • GraphProduct is also known as box product.
  • GraphProduct is typically used to produce new graphs from Boolean combinations of initial graphs.
  • GraphProduct[g1,g2] gives a graph with vertices formed from the Cartesian product of the vertices of g1 and vertices of g2. The vertices {u1,u2} and {v1,v2} are connected if u1v1 and u2 is connected to v2, or u2v2 and u1 is connected to v1.
  • GraphProduct[g1,g2,"op"] gives a graph product of type "op" with edges {u1,u2}{v1,v2} subject to the following conditions:
  • "Cartesian"(u1v1 ∧ u2v2)∨(u2v2∧u1v1)
    "Conormal"(u1v1)∨(u2v2)
    "Lexicographical"(u1v1)∨(u1v1∧u2v2)
    "Normal"(u1v1∧u2v2)∨(u2v2∧u1v1)∨(u1v1∧u2v2)
    "Rooted"(u1v1 ∧ u2v2)∨(u1v1 ∧ u2v2r)
    "Tensor"(u1v1)∧(u2v2)
  • The vertex r is the first vertex in VertexList[g2].
  • GraphProduct[g1,g2] is effectively equivalent to GraphProduct[g1,g2,"Cartesian"].
  • GraphProduct works with undirected graphs, directed graphs, multigraphs and mixed graphs.

Examples

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Basic Examples  (3)

Cartesian product of two graphs:

A table of graph products:

Generate grid graphs:

Torus graphs:

Scope  (30)

Directed Graphs  (5)

GraphProduct works with directed graphs:

Simple directed graphs:

Directed multigraphs:

Directed weighted graphs:

Directed annotated graphs:

Undirected Graphs  (5)

GraphProduct works with undirected graphs:

Simple undirected graphs:

Undirected multigraphs:

Undirected weighted graphs:

Undirected annotated graphs:

Mixed Graphs  (5)

GraphProduct works with mixed graphs:

Simple mixed graphs:

Mixed multigraphs:

Mixed weighted graphs:

Mixed annotated graphs:

Multigraphs  (5)

GraphProduct works with multigraphs:

Directed multigraphs:

Mixed multigraphs:

Weighted multigraphs:

Annotated multigraphs:

Weighted Graphs  (5)

GraphProduct works with weighted graphs:

Directed weighted graphs:

Undirected weighted graphs:

Mixed weighted graphs:

Annotated weighted graphs:

Special Graphs  (5)

GraphProduct works on entity graphs:

GraphProduct works on trees:

Use rules to specify the graph:

GraphProduct works with more than two graphs:

Generate a list of different graph products:

Properties & Relations  (6)

For two graphs with vi vertices, the number of vertices of their product is v1 v2 :

For two undirected graphs with vi vertices and ei edges, the number of edges of the Cartesian product is v1 e2+v2 e1:

Tensor product is 2 e1e2:

Lexicographical product is v1 e2+ e1v22 :

Normal product is v1 e2+v2 e1 + 2 e1e2:

Co-normal product is v12 e2+ e1v22 - 2e1e2:

Rooted product is v1 e2+ e1:

The Cartesian product of two single edges is a cycle:

The normal product of two single edges is a complete graph:

The tensor product of two single edges is a cross:

TorusGraph[{m,n}] is the graph formed from the Cartesian product of the cycle graphs and :

Wolfram Research (2022), GraphProduct, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphProduct.html.

Text

Wolfram Research (2022), GraphProduct, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphProduct.html.

CMS

Wolfram Language. 2022. "GraphProduct." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/GraphProduct.html.

APA

Wolfram Language. (2022). GraphProduct. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GraphProduct.html

BibTeX

@misc{reference.wolfram_2025_graphproduct, author="Wolfram Research", title="{GraphProduct}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/GraphProduct.html}", note=[Accessed: 01-December-2025]}

BibLaTeX

@online{reference.wolfram_2025_graphproduct, organization={Wolfram Research}, title={GraphProduct}, year={2022}, url={https://reference.wolfram.com/language/ref/GraphProduct.html}, note=[Accessed: 01-December-2025]}