Edge-transitive graph

In mathematics, 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

Edge-transitive graphs include any complete bipartite graph $K_{m,n}$ , 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.

References

^ ^a ^b ^c Biggs, Norman (1993). Algebraic Graph Theory (2nd ed. ed.). Cambridge: Cambridge University Press. p. 118. ISBN 0-521-45897-8. {{cite book}}: |edition= has extra text (help)

External links

Weisstein, Eric W. "Edge-transitive graph". MathWorld.

[biggs-1] Biggs, Norman (1993). Algebraic Graph Theory (2nd ed. ed.). Cambridge: Cambridge University Press. p. 118. ISBN 0-521-45897-8. {{cite book}}: |edition= has extra text (help)

[1]

Graph families defined by their automorphisms
distance-transitive	→	distance-regular	←	strongly regular
↓
symmetric (arc-transitive)	←	t-transitive, $t \geq 2$		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

Examples and properties

See also

References

External links