Let $\mathcal G_n$ be the set of (isomorphism classes of unlabelled) simple graphs on $n$ vertices.

I wonder whether there are any 'interesting' combinatorial parameters $a,b: \mathcal G_n\to \mathbb N$, which are conjecturally equidistributed, that is, $$ \sum_{G\in\mathcal G_n} q^{a(G)} = \sum_{G\in\mathcal G_n} q^{b(G)}. $$

I would also be interested in such parameters where the proof is not entirely straightforward.