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VertexTransitiveGraphQ
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VertexTransitiveGraphQ

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yields True if the graph g is a vertex–transitive graph and False otherwise.

Details

  • A graph g is vertex transitive if for any vertices v and w of g, there is an automorphism of g that maps v to w.
  • VertexTransitiveGraphQ is typically used to test whether all vertices in a graph have identical neighborhoods.

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

Test whether a graph is vertex transitive:

In[1]:=1
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-360cz6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-w9jqor
Out[2]=2

A star graph is not vertex transitive:

In[1]:=1
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-w2fx9b
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-i6ltge
Out[2]=2

Scope  (7)Survey of the scope of standard use cases

Test undirected graphs:

In[1]:=1
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-8zzfce
Out[1]=1

Directed graphs:

In[1]:=1
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-xkbvj
Out[1]=1

Multigraphs:

In[1]:=1
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-yyp8c5
Out[1]=1

Mixed graphs:

In[1]:=1
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-xb9nxo
Out[1]=1

Tagged graphs:

In[1]:=1
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-m857vb
Out[1]=1

VertexTransitiveGraphQ gives False for anything that is not a vertex–transitive graph:

In[1]:=1
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-3l2bwe
Out[1]=1

VertexTransitiveGraphQ works with large graphs:

In[1]:=1
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https://wolfram.com/xid/0cf3zahcbbp2xqb-pq9ae
In[2]:=2
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https://wolfram.com/xid/0cf3zahcbbp2xqb-cevvx1
Out[2]=2

Applications  (1)Sample problems that can be solved with this function

Generate a list of vertex–transitive graphs from GraphData:

In[1]:=1
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https://wolfram.com/xid/0cf3zahcbbp2xqb-603g5m

Check:

In[2]:=2
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https://wolfram.com/xid/0cf3zahcbbp2xqb-ed4ahe
Out[2]=2

Properties & Relations  (7)Properties of the function, and connections to other functions

Every vertex–transitive graph is regular:

In[4]:=4
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https://wolfram.com/xid/0cf3zahcbbp2xqb-gs954j
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-jikebi
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-gpcx16
Out[6]=6

The graph complement of a vertex–transitive graph is vertex transitive:

In[1]:=1
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-xod50k
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-1kw7xn
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0cf3zahcbbp2xqb-fspuji
Out[3]=3

Use GraphAutomorphismGroup to test whether a graph is vertex transitive:

In[1]:=1
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-pzifs3
In[2]:=2
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-xcrc61
Out[2]=2

Find the automorphism group:

In[3]:=3
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https://wolfram.com/xid/0cf3zahcbbp2xqb-tfr4y4

Compute the orbit of a permutation group:

In[4]:=4
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https://wolfram.com/xid/0cf3zahcbbp2xqb-ihx7k3
Out[4]=4

Single orbit should permute all vertices:

In[5]:=5
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https://wolfram.com/xid/0cf3zahcbbp2xqb-hxmzx5
Out[5]=5

Use VertexTransitiveGraphQ to test whether a connected graph is edge transitive:

In[1]:=1
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-ntemc3
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-yttdy2
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0cf3zahcbbp2xqb-8q98dx
Out[3]=3

The edge connectivity of a vertex-transitive graph is equal to the degree :

In[1]:=1
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-35kbp7
In[2]:=2
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-kvnisa
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-x1qmql
Out[3]=3

The vertex connectivity of a vertex-transitive graph will be at least :

In[1]:=1
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-bgd5tj
In[2]:=2
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-sn6gl6
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-s5bje1
Out[3]=3

The vertex-transitive graph includes CompleteGraph:

In[1]:=1
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-cwslkt
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-169mmj
Out[2]=2

CycleGraph:

In[3]:=3
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-v9a0f6
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-osk1pp
Out[4]=4

PetersenGraph:

In[5]:=5
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-o0m9k5
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-codql8
Out[6]=6

Heawood graph:

In[7]:=7
✖
https://wolfram.com/xid/0cf3zahcbbp2xqb-bwy8b1
Out[7]=7
In[8]:=8
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https://wolfram.com/xid/0cf3zahcbbp2xqb-wphwxi
Out[8]=8
Wolfram Research (2021), VertexTransitiveGraphQ, Wolfram Language function, https://reference.wolfram.com/language/ref/VertexTransitiveGraphQ.html.
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Wolfram Research (2021), VertexTransitiveGraphQ, Wolfram Language function, https://reference.wolfram.com/language/ref/VertexTransitiveGraphQ.html.

Text

Wolfram Research (2021), VertexTransitiveGraphQ, Wolfram Language function, https://reference.wolfram.com/language/ref/VertexTransitiveGraphQ.html.

✖
Wolfram Research (2021), VertexTransitiveGraphQ, Wolfram Language function, https://reference.wolfram.com/language/ref/VertexTransitiveGraphQ.html.

CMS

Wolfram Language. 2021. "VertexTransitiveGraphQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/VertexTransitiveGraphQ.html.

✖
Wolfram Language. 2021. "VertexTransitiveGraphQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/VertexTransitiveGraphQ.html.

APA

Wolfram Language. (2021). VertexTransitiveGraphQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/VertexTransitiveGraphQ.html

✖
Wolfram Language. (2021). VertexTransitiveGraphQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/VertexTransitiveGraphQ.html

BibTeX

@misc{reference.wolfram_2025_vertextransitivegraphq, author="Wolfram Research", title="{VertexTransitiveGraphQ}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/VertexTransitiveGraphQ.html}", note=[Accessed: 26-March-2025 ]}

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@misc{reference.wolfram_2025_vertextransitivegraphq, author="Wolfram Research", title="{VertexTransitiveGraphQ}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/VertexTransitiveGraphQ.html}", note=[Accessed: 26-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_vertextransitivegraphq, organization={Wolfram Research}, title={VertexTransitiveGraphQ}, year={2021}, url={https://reference.wolfram.com/language/ref/VertexTransitiveGraphQ.html}, note=[Accessed: 26-March-2025 ]}

✖
@online{reference.wolfram_2025_vertextransitivegraphq, organization={Wolfram Research}, title={VertexTransitiveGraphQ}, year={2021}, url={https://reference.wolfram.com/language/ref/VertexTransitiveGraphQ.html}, note=[Accessed: 26-March-2025 ]}