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% RequiredMacros: \DeclareMathOperator{\aut}{Aut} 
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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 $\aut(\mathcal{G})$.