# Edge-transitive graph

 Graph families defined by their automorphisms distance-transitive ${\displaystyle {\boldsymbol {\rightarrow }}}$ distance-regular ${\displaystyle {\boldsymbol {\leftarrow }}}$ strongly regular ${\displaystyle {\boldsymbol {\downarrow }}}$ symmetric (arc-transitive) ${\displaystyle {\boldsymbol {\leftarrow }}}$ t-transitive, t ≥ 2 skew-symmetric ${\displaystyle {\boldsymbol {\downarrow }}}$ (if connected) vertex- and edge-transitive ${\displaystyle {\boldsymbol {\rightarrow }}}$ edge-transitive and regular ${\displaystyle {\boldsymbol {\rightarrow }}}$ edge-transitive ${\displaystyle {\boldsymbol {\downarrow }}}$ ${\displaystyle {\boldsymbol {\downarrow }}}$ ${\displaystyle {\boldsymbol {\downarrow }}}$ vertex-transitive ${\displaystyle {\boldsymbol {\rightarrow }}}$ regular ${\displaystyle {\boldsymbol {\rightarrow }}}$ (if bipartite) biregular ${\displaystyle {\boldsymbol {\uparrow }}}$ Cayley graph ${\displaystyle {\boldsymbol {\leftarrow }}}$ 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 upon its edges.

## Examples and properties

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

Edge-transitive graphs include any complete bipartite graph ${\displaystyle K_{m,n}}$, and any 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. The Gray graph is an example of a graph which is edge-transitive but not vertex-transitive. All such graphs are bipartite,[1] and hence can be colored with only two colors.

An edge-transitive graph that is also regular, but not vertex-transitive, is called semi-symmetric. The Gray graph again provides an example. Every edge-transitive graph that is not vertex-transitive must be bipartite and either semi-symmetric or biregular.[2]

## References

1. ^ a b c 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.