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Edge-transitive graph

 

 
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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, an edge-transitive graph is a graph
Graph (mathematics)

In mathematics a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges....
 G such that, given any two edges e1 and e2 of G, there is an






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, an edge-transitive graph is a graph
Graph (mathematics)

In mathematics a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges....
 G such that, given any two edges e1 and e2 of G, there is an automorphism
Graph automorphism

In graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge?vertex connectivity....
 of G that maps e1 to e2.

In other words, a graph is edge-transitive if its automorphism group acts transitively
Group action

In algebra and geometry, a group action is a way of describing symmetry of objects using group . The essential elements of the object are described by a Set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformation of the set....
 upon its edges.

Examples and properties

  • Any complete bipartite graph
    Complete bipartite graph

    In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set....
      is edge-transitive.
  • Any edge-transitive graph that is not vertex-transitive is bipartite. These graphs are called semi-symmetric
    Semi-symmetric graph

    In mathematics, a semi-symmetric graph is an undirected graph that is edge-transitive graph and regular graph, but not Vertex-transitive graph....
    .


See also

  • Vertex-transitive graph
    Vertex-transitive graph

    In mathematics, a vertex-transitive graph is a Graph G such that, given any two vertices v1 and v2 of G, there is some Graph automorphism...
  • Arc-transitive graph
    Arc-transitive graph

    In mathematics, an arc-transitive graph is a graph G such that, given any two edges e1 = u1v1 and e2 = u2v2 of G, there are two Graph automorphisms...
  • Edge-transitive (in geometry)


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