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I'm reading some old article and I have one small question: what in general is the group of a graph? By the article, definition should be in Harary's Graph Theory, but unfortunately I don't have any access to that book.

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The group of $G$ just means the automorphism group of $G$, i.e. the collection of all isomorphisms of $G$ with itself, the group operation being composition.

Here is the relevant section from Harary's Graph Theory:

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