An is a permutation of the set of vertices such that two vertices and form an edge if and only if and form an edge. The is the set of all automorphisms of the graph. If is a graph, its automorphism group is denoted .
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An \textdef{automorphism of a graph} is a permutation $\sigma$ of the set of vertices such that two vertices $x$ and $y$ form an edge if and only if $\sigma(x)$ and $\sigma(y)$ form an edge. The \textdef{automorphism group of a graph} is the set of all automorphisms of the graph. If $\mathcal{G}$ is a graph, its automorphism group is denoted $\aut(\mathcal{G})$.