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High Quality Content by WIKIPEDIA articles! In the mathematical field of graph theory, a graph is symmetric if its automorphism group acts transitively upon ordered pairs of linked vertices (that is, upon edges considered as having a direction). Such a graph is sometimes also called 1-arc-transitive or flag-transitive. By definition (ignoring u1 and u2), 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. Semi-symmetric graphs, for example, are edge-transitive and regular, but not vertex-transitive.