Having already been through this issue with tensor products, I have no doubt that graph theorists make up their own definitions for standard mathematical objects. But if you claim there is more than one cyclic group of order 6, that's not something I have to argue, any more than if you said the sun rises in the west and dared me to prove you wrong. I don't need to even have the conversation. You're just wrong on this. Are you honestly claiming that \(\mathbb Z_2 \oplus \mathbb Z_3\) is not isomorphic to \(\mathbb Z_6\)? How can you say that and expect to be taken seriously? You might as well claim that 1 + 1 is 3.