Edge-transitive graph

From HandWiki
Short description: Graph where all pairs of edges are automorphic
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 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]

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 [math]\displaystyle{ K_{m,n} }[/math] where m ≠ n, which includes the star graphs [math]\displaystyle{ K_{1,n} }[/math]. 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

  • Edge-transitive (in geometry)

References

  1. 1.0 1.1 1.2 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, https://books.google.com/books?id=hsymFm0E0uIC&pg=PA20 .
  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. doi:10.2140/involve.2019.12.1329. 
  4. Watkins, Mark E. (1970), "Connectivity of transitive graphs", Journal of Combinatorial Theory 8: 23–29, doi:10.1016/S0021-9800(70)80005-9 

External links