VertexTransitiveGraphQ

VertexTransitiveGraphQ[g]

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

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Basic Examples  (2)

Test whether a graph is vertex transitive:

A star graph is not vertex transitive:

Scope  (7)

Test undirected graphs:

Directed graphs:

Multigraphs:

Mixed graphs:

Tagged graphs:

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

VertexTransitiveGraphQ works with large graphs:

Applications  (1)

Generate a list of vertex–transitive graphs from GraphData:

Check:

Properties & Relations  (7)

Every vertex–transitive graph is regular:

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

Use GraphAutomorphismGroup to test whether a graph is vertex transitive:

Find the automorphism group:

Compute the orbit of a permutation group:

Single orbit should permute all vertices:

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

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

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

The vertex-transitive graph includes CompleteGraph:

CycleGraph:

PetersenGraph:

Heawood graph:

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.

CMS

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

BibTeX

@misc{reference.wolfram_2023_vertextransitivegraphq, author="Wolfram Research", title="{VertexTransitiveGraphQ}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/VertexTransitiveGraphQ.html}", note=[Accessed: 19-April-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_vertextransitivegraphq, organization={Wolfram Research}, title={VertexTransitiveGraphQ}, year={2021}, url={https://reference.wolfram.com/language/ref/VertexTransitiveGraphQ.html}, note=[Accessed: 19-April-2024 ]}