Edge-transitive graph

In the mathematical field of graph theory, an edge-transitive graph is a graph G such that, given any two edges e1 and e2 of G, there is an automorphism of G that maps e1 to e2.[1]

This article is about graph theory. For edge transitivity in geometry, see Edge-transitive.
Graph families defined by their automorphisms
distance-transitive → distance-regular ← strongly regular
↓
symmetric (arc-transitive) ← t-transitive, t ≥ 2 skew-symmetric
↓
(if connected)
vertex- and edge-transitive		 → edge-transitive and regular → edge-transitive
↓ ↓ ↓
vertex-transitive → regular → (if bipartite)
biregular
↑
Cayley graph ← zero-symmetric asymmetric

In other words, a graph is edge-transitive if its automorphism group acts transitively on its edges.

Examples and properties
The Gray graph is edge-transitive and regular, but not vertex-transitive.

The number of connected simple edge-transitive graphs on n vertices is 1, 1, 2, 3, 4, 6, 5, 8, 9, 13, 7, 19, 10, 16, 25, 26, 12, 28 ... (sequence A095424 in the OEIS)

Edge-transitive graphs include all symmetric graph, such as the vertices and edges of the cube.[1] Symmetric graphs are also vertex-transitive (if they are connected), but in general edge-transitive graphs need not be vertex-transitive. Every connected edge-transitive graph that is not vertex-transitive must be bipartite,[1] (and hence can be colored with only two colors), and either semi-symmetric or biregular.[2]

Examples of edge but not vertex transitive graphs include the complete bipartite graphs where m ≠ n, which includes the star graphs . For graphs on n vertices, there are (n-1)/2 such graphs for odd n and (n-2) for even n. Additional edge transitive graphs which are not symmetric can be formed as subgraphs of these complete bi-partite graphs in certain cases. Subgraphs of complete bipartite graphs Km,n exist when m and n share a factor greater than 2. When the greatest common factor is 2, subgraphs exist when 2n/m is even or if m=4 and n is an odd multiple of 6.[3] So edge transitive subgraphs exist for K3,6, K4,6 and K5,10 but not K4,10. An alternative construction for some edge transitive graphs is to add vertices to the midpoints of edges of a symmetric graph with v vertices and e edges, creating a bipartite graph with e vertices of order 2, and v of order 2e/v.

An edge-transitive graph that is also regular, but still not vertex-transitive, is called semi-symmetric. The Gray graph, a cubic graph on 54 vertices, is an example of a regular graph which is edge-transitive but not vertex-transitive. The Folkman graph, a quartic graph on 20 vertices is the smallest such graph.

The vertex connectivity of an edge-transitive graph always equals its minimum degree.[4]

See also

References
  1. Biggs, Norman (1993). Algebraic Graph Theory (2nd ed.). Cambridge: Cambridge University Press. p. 118. ISBN 0-521-45897-8.
  2. Lauri, Josef; Scapellato, Raffaele (2003), Topics in Graph Automorphisms and Reconstruction, London Mathematical Society Student Texts, Cambridge University Press, pp. 20–21, ISBN 9780521529037.
  3. Newman, Heather A.; Miranda, Hector; Narayan, Darren A (2019). "Edge-transitive graphs and combinatorial designs". Involve: A Journal of Mathematics. 12 (8): 1329–1341. arXiv:1709.04750. doi:10.2140/involve.2019.12.1329. S2CID 119686233.
  4. Watkins, Mark E. (1970), "Connectivity of transitive graphs", Journal of Combinatorial Theory, 8: 23–29, doi:10.1016/S0021-9800(70)80005-9, MR 0266804
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