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# Edge-Transitive Graph

An edge-transitive graph is a graph such that any two edges are equivalent under some element of its automorphism group. More precisely, a graph is edge-transitive if for all pairs of edges there exists an element of the edge automorphism group such that (Holton and Sheehan 1993, p. 28). Informally speaking, a graph is edge-transitive if every edge has the same local environment, so that no edge can be distinguished from any other based on the vertices and edges surrounding it.

By convention, the singleton graph and 2-path graph are considered edge-transitive (B. McKay, pers. comm., Mar. 22, 2007).

A graph may be tested to determine if it is edge-transitive in the Wolfram Language using EdgeTransitiveGraphQ[g].

A connected undirected graph is edge-transitive iff its line graph is vertex-transitive. Note that this statement does not hold in general for disconnected graphs, since for example the line graph of the disjoint graph union of the cycle graph and the claw graph is isomorphic to the graph disjoint union of two triangle graphs, which is edge-transitive, while the original graph is clearly not.

Counting empty graphs as edge-transitive, the numbers of edge-transitive graphs on , 2, ... nodes are 1, 2, 4, 8, 12, 21, 27, 39, 50, 69, ... (OEIS A095352).

The numbers of connected edge-transitive graphs on , 2, ... nodes are 1, 1, 2, 3, 4, 6, 5, 8, 9, 13, 7, ... (OEIS A095424; c.f. Conder 2017).

A graph that is both edge-transitive and vertex-transitive is called a symmetric graph (Holton and Sheehan 1993, pp. 209-210). A regular graph that is edge-transitive but not vertex-transitive is called a semisymmetric graph.

Arc-Transitive Graph, Automorphism Group, Edge Automorphism, Edge Automorphism Group, Gray Graph, Folkman Graph, Semisymmetric Graph, Symmetric Graph, Vertex-Transitive Graph

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## References

Condor, M. "All Connected Edge-Transitive Graphs on Up to 47 Vertices." 24 Jul 2017. https://www.math.auckland.ac.nz/~conder/AllSmallETgraphs-upto47-summary.txt.Holton, D. A. and Sheehan, J. The Petersen Graph. Cambridge, England: Cambridge University Press, 1993.Lauri, J. and Scapellato, R. Topics in Graph Automorphisms and Reconstruction. Cambridge, England: Cambridge University Press, 2003.Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990.Sloane, N. J. A. Sequences A095352 and A095424 in "The On-Line Encyclopedia of Integer Sequences."

## Referenced on Wolfram|Alpha

Edge-Transitive Graph

## Cite this as:

Weisstein, Eric W. "Edge-Transitive Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Edge-TransitiveGraph.html