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EdgeTransitiveGraphQ—Wolfram Documentation

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EdgeTransitiveGraphQ[g]

yields True if the graph g is a edge-transitive graph and False otherwise.

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EdgeTransitiveGraphQ

EdgeTransitiveGraphQ[g]

yields True if the graph g is a edge-transitive graph and False otherwise.

Details

  • A graph g is edge transitive if for any edges e1 and e2 of g, there is an automorphism of g that maps e1 to e2.
  • EdgeTransitiveGraphQ is typically used to test whether all edges in a graph have identical neighborhoods.

Examples

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Basic Examples  (2)

Test whether a graph is edge transitive:

Wolfram Language code: CompleteGraph[3]
Wolfram Language code: EdgeTransitiveGraphQ[%]

A wheel graph is not edge transitive:

Wolfram Language code: WheelGraph[5]
Wolfram Language code: EdgeTransitiveGraphQ[%]

Scope  (7)

Test undirected graphs:

Wolfram Language code: EdgeTransitiveGraphQ[[image]]

Directed graphs:

Wolfram Language code: EdgeTransitiveGraphQ[[image]]

Multigraphs:

Wolfram Language code: EdgeTransitiveGraphQ[[image]]

Mixed graphs:

Wolfram Language code: EdgeTransitiveGraphQ[[image]]

Tagged graphs:

Wolfram Language code: EdgeTransitiveGraphQ[[image]]

EdgeTransitiveGraphQ gives False for anything that is not an edge-transitive graph:

Wolfram Language code: EdgeTransitiveGraphQ[a]

EdgeTransitiveGraphQ works with large graphs:

Wolfram Language code: GridGraph[{10, 10, 10, 10}];
Wolfram Language code: EdgeTransitiveGraphQ[%]//Timing

Applications  (1)

Generate a list of edge-transitive graphs from GraphData:

Wolfram Language code: GraphData["EdgeTransitive"]//Short

Check:

Wolfram Language code: EdgeTransitiveGraphQ[GraphData[#]] & /@ Take[%, 5]

Properties & Relations  (5)

Connected edge-transitive graphs are either vertex transitive or bipartite:

Wolfram Language code: data = Intersection[GraphData["EdgeTransitive"], GraphData["Connected"]]; graphs = GraphData[#] & /@ Take[data, {6, 11}]
Wolfram Language code: (VertexTransitiveGraphQ[#] || BipartiteGraphQ[#])& /@ graphs

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

Wolfram Language code: g = GraphData[{"Arrangement", {5, 2}}]
Wolfram Language code: VertexTransitiveGraphQ[LineGraph[g]]
Wolfram Language code: EdgeTransitiveGraphQ[g]

THe edge-transitive graphs need not be vertex transitive:

Wolfram Language code: g = GraphData["GrayGraph"]
Wolfram Language code: EdgeTransitiveGraphQ[g]
Wolfram Language code: VertexTransitiveGraphQ[g]

The vertex connectivity of an edge-transitive graph equals its minimum degree:

Wolfram Language code: g = [image];
Wolfram Language code: EdgeTransitiveGraphQ[g]
Wolfram Language code: VertexConnectivity[g] == Min[VertexDegree[g]]

The edge-transitive graph includes CompleteGraph:

Wolfram Language code: Table[CompleteGraph[n], {n, 3, 6}]
Wolfram Language code: EdgeTransitiveGraphQ /@ %

Complete bipartite graph:

Wolfram Language code: Table[CompleteGraph[{n, n}], {n, 3, 6}]

CycleGraph:

Wolfram Language code: Table[CycleGraph[n], {n, 3, 6}]
Wolfram Language code: EdgeTransitiveGraphQ /@ %

Gray graph:

Wolfram Language code: GraphData["GrayGraph"]
Wolfram Language code: EdgeTransitiveGraphQ[%]
Wolfram Research (2021), EdgeTransitiveGraphQ, Wolfram Language function, https://reference.wolfram.com/language/ref/EdgeTransitiveGraphQ.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_edgetransitivegraphq, author="Wolfram Research", title="{EdgeTransitiveGraphQ}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/EdgeTransitiveGraphQ.html}", note=[Accessed: 14-November-2026]}

BibLaTeX

@online{reference.wolfram_2026_edgetransitivegraphq, organization={Wolfram Research}, title={EdgeTransitiveGraphQ}, year={2021}, url={https://reference.wolfram.com/language/ref/EdgeTransitiveGraphQ.html}, note=[Accessed: 14-November-2026]}