An edge-transitive graph is a graph such that any two edges are equivalent under some element of its automorphism group. More precisely, a graph is edge-transitive if for all pairs of edges (e_1, e_2) there exists an element γ of the edge automorphism group Aut^*(G) such that γ(e_1) = e_2. Informally speaking, a graph is edge-transitive if every edge has the same local environment, so that no edge can be distinguished from any other based on the vertices and edges surrounding it. By convention, the singleton graph and 2-path graph are considered edge-transitive. A graph may be tested to determine if it is edge-transitive in the Wolfram Language using EdgeTransitiveGraphQ[g].
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