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$\mathrm{with}\left(\mathrm{GraphTheory}\right)\:$$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

Compute the automorphism group of the cycle graph on 5 vertices and verify it is isomorphic to the dihedral group D5.
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$\mathrm{C5}\u2254\mathrm{CycleGraph}\left(5\right)$

${\mathrm{C5}}{\u2254}{\mathrm{Graph\; 1:\; an\; undirected\; unweighted\; graph\; with\; 5\; vertices\; and\; 5\; edge(s)}}$
 (1) 
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$G\u2254\mathrm{AutomorphismGroup}\left(\mathrm{C5}\right)$

${G}{\u2254}\u27e8\left({1}{\,}{2}\right)\left({3}{\,}{5}\right){\,}\left({2}{\,}{5}\right)\left({3}{\,}{4}\right)\u27e9$
 (2) 
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$\mathrm{AreIsomorphic}\left(G\,\mathrm{DihedralGroup}\left(5\right)\right)$

Compute the automorphism group of the complete graph on 4 vertices and verify it is isomorphic to the symmetric group S4.
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$\mathrm{K4}\u2254\mathrm{CompleteGraph}\left(4\right)$

${\mathrm{K4}}{\u2254}{\mathrm{Graph\; 2:\; an\; undirected\; unweighted\; graph\; with\; 4\; vertices\; and\; 6\; edge(s)}}$
 (4) 
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$G\u2254\mathrm{AutomorphismGroup}\left(\mathrm{K4}\right)$

${G}{\u2254}\u27e8\left({3}{\,}{4}\right){\,}\left({1}{\,}{2}\right){\,}\left({2}{\,}{3}\right)\u27e9$
 (5) 
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$\mathrm{AreIsomorphic}\left(G\,\mathrm{SymmetricGroup}\left(4\right)\right)$

Compute the automorphism group of the Petersen graph and display its order.
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$\mathrm{PG}\u2254\mathrm{SpecialGraphs}:\mathrm{PetersenGraph}\left(\right)$

${\mathrm{PG}}{\u2254}{\mathrm{Graph\; 3:\; an\; undirected\; unweighted\; graph\; with\; 10\; vertices\; and\; 15\; edge(s)}}$
 (7) 
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$G\u2254\mathrm{AutomorphismGroup}\left(\mathrm{PG}\right)$

${G}{\u2254}\u27e8\left({1}{\,}{2}\right)\left({3}{\,}{5}\right)\left({6}{\,}{9}\right)\left({7}{\,}{8}\right){\,}\left({2}{\,}{5}\right)\left({3}{\,}{4}\right)\left({7}{\,}{10}\right)\left({8}{\,}{9}\right){\,}\left({3}{\,}{9}\right)\left({4}{\,}{8}\right)\left({7}{\,}{10}\right){\,}\left({4}{\,}{7}\right)\left({5}{\,}{6}\right)\left({8}{\,}{10}\right)\u27e9$
 (8) 
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$\mathrm{GroupOrder}\left(G\right)$
