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Because the number of possible variations explodes when multiple edges are allowed, you're unlikely to find much (if any) previous research on such a specific class as you've defined.
House of Graphs (hog.grinvin.org) can occasionally help you with this sort of question, but it is not nearly as large or well-established as the OEIS and it does not deal with multigraphs.
You can check the equation modulo $n$ for various $n$. For example, there are no integer solutions when $a \equiv 0 \!\!\!\mod \!3$ and $b \equiv 2 \!\!\!\mod \!3$.
