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In the mathematical field of graph theory, a graph G is symmetric if, given any two pairs of adjacent vertices u1—v1 and u2—v2 of G, there is an automorphism
such that
In other words, a graph is symmetric if its automorphism group acts transitively on ordered pairs of adjacent vertices. Such a graph is sometimes also called 1-arc-transitive or flag-transitive.
By definition , a symmetric graph without isolated vertices must also be vertex-transitive. Since the definition above maps one edge to another, a symmetric graph must also be edge-transitive. However, an edge-transitive graph need not be symmetric, since a—b might map to c—d, but not to d—c. Star graphs are a simple example of being edge-transitive without being vertex-transitive or symmetric. As a further example, semi-symmetric graphs are edge-transitive and regular, but not vertex-transitive.