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.
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: