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GraphProduct
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GraphProduct

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GraphProduct[g1,g2]

gives the Cartesian product of the graphs g1 and g2.

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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

open allclose all

Basic Examples  (3)Summary of the most common use cases

Cartesian product of two graphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-evj2e0
Out[1]=1

A table of graph products:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-8utcrx
Out[1]=1

Generate grid graphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-jih14k
Out[1]=1

Torus graphs:

In[2]:=2
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https://wolfram.com/xid/0cg5b7xd2-28e2uk
Out[2]=2

Scope  (30)Survey of the scope of standard use cases

Directed Graphs  (5)

GraphProduct works with directed graphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-zq47nk
Out[1]=1

Simple directed graphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-5opcl
Out[1]=1

Directed multigraphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-lb4tx5
Out[1]=1

Directed weighted graphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-5komsb
Out[1]=1

Directed annotated graphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-wpl3zt
Out[1]=1

Undirected Graphs  (5)

GraphProduct works with undirected graphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-uv48kz
Out[1]=1

Simple undirected graphs:

In[1]:=1
✖
https://wolfram.com/xid/0cg5b7xd2-hf3no2
Out[1]=1

Undirected multigraphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-c1omjq
Out[1]=1

Undirected weighted graphs:

In[1]:=1
✖
https://wolfram.com/xid/0cg5b7xd2-yvj1t
Out[1]=1

Undirected annotated graphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-0zhqif
Out[1]=1

Mixed Graphs  (5)

GraphProduct works with mixed graphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-xxc6u3
Out[1]=1

Simple mixed graphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-lrsx54
Out[1]=1

Mixed multigraphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-4o5kuy
Out[1]=1

Mixed weighted graphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-cz38bv
Out[1]=1

Mixed annotated graphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-urfdvp
Out[1]=1

Multigraphs  (5)

GraphProduct works with multigraphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-tsn0f4
Out[1]=1

Directed multigraphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-ssgm65
Out[1]=1

Mixed multigraphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-t1xhwo
Out[1]=1

Weighted multigraphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-expkz4
Out[1]=1

Annotated multigraphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-2d0zdg
Out[1]=1

Weighted Graphs  (5)

GraphProduct works with weighted graphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-7csf8k
Out[1]=1

Directed weighted graphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-jpcxhu
Out[1]=1

Undirected weighted graphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-z0p727
Out[1]=1

Mixed weighted graphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-rkv9ly
Out[1]=1

Annotated weighted graphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-jl91ra
Out[1]=1

Special Graphs  (5)

GraphProduct works on entity graphs:

In[2]:=2
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https://wolfram.com/xid/0cg5b7xd2-9q5ywu
Out[2]=2

GraphProduct works on trees:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-3nrjxt
Out[1]=1

Use rules to specify the graph:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-weyyix
Out[1]=1

GraphProduct works with more than two graphs:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-18b1vu
Out[1]=1

Generate a list of different graph products:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-vt8dmp
In[2]:=2
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https://wolfram.com/xid/0cg5b7xd2-b5h9xv
Out[2]=2

Properties & Relations  (6)Properties of the function, and connections to other functions

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

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-vf4rqa
In[2]:=2
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https://wolfram.com/xid/0cg5b7xd2-r610af
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-1v9tiq
In[2]:=2
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https://wolfram.com/xid/0cg5b7xd2-s4nwjo
In[3]:=3
✖
https://wolfram.com/xid/0cg5b7xd2-owlx2z
In[4]:=4
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https://wolfram.com/xid/0cg5b7xd2-u1y6x8
Out[4]=4

Tensor product is 2 e1e2:

In[5]:=5
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https://wolfram.com/xid/0cg5b7xd2-rgscwb
Out[5]=5

Lexicographical product is v1 e2+ e1v22 :

In[6]:=6
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https://wolfram.com/xid/0cg5b7xd2-3xis6p
Out[6]=6

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

In[7]:=7
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https://wolfram.com/xid/0cg5b7xd2-sj10w7
Out[7]=7

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

In[8]:=8
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https://wolfram.com/xid/0cg5b7xd2-ohf7yr
Out[8]=8

Rooted product is v1 e2+ e1:

In[9]:=9
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https://wolfram.com/xid/0cg5b7xd2-2lxl2c
Out[9]=9

The Cartesian product of two single edges is a cycle:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-xrbynk
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0cg5b7xd2-fe7e69
Out[1]=1

The tensor product of two single edges is a cross:

In[1]:=1
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https://wolfram.com/xid/0cg5b7xd2-z10r6r
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0cg5b7xd2-2xqjjb
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cg5b7xd2-l8vjru
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cg5b7xd2-irxwob
Out[3]=3
Wolfram Research (2022), GraphProduct, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphProduct.html.
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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.

✖
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.

✖
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

✖
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: 08-July-2025 ]}

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@misc{reference.wolfram_2025_graphproduct, author="Wolfram Research", title="{GraphProduct}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/GraphProduct.html}", note=[Accessed: 08-July-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: 08-July-2025 ]}

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@online{reference.wolfram_2025_graphproduct, organization={Wolfram Research}, title={GraphProduct}, year={2022}, url={https://reference.wolfram.com/language/ref/GraphProduct.html}, note=[Accessed: 08-July-2025 ]}