Edge transitive graph - Alchetron, The Free Social Encyclopedia

→ {\displaystyle {\boldsymbol {\rightarrow }}} distance-regular ← {\displaystyle {\boldsymbol {\leftarrow }}} t-transitive, t ≥ 2 → {\displaystyle {\boldsymbol {\rightarrow }}} regular		← {\displaystyle {\boldsymbol {\leftarrow }}} strongly regular → {\displaystyle {\boldsymbol {\rightarrow }}} edge-transitive

→ {\displaystyle {\boldsymbol {\rightarrow }}} edge-transitive and regular

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.

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. 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, 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.

References

Edge-transitive graph Wikipedia

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