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@LouisD I've tried to clarify the question. I'm interested in a set of labels that ensures, no matter how you arrange the labels on the $K_n$, that the network will still be irregular. That is, the labels should work for all assignments, not just one. Sorry for the poor wording before.
And thank you for making an answer. But I wasn't thinking, and the quantifier is wrong: I want a set that guarantees an irregular network, not one that just permits it. I at least have a starting point though, so if I get anywhere from the lead you gave, I'll accept the answer with a comment.
