equidistributed parameters on graphs

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.

As a spin-off of https://mathoverflow.net/a/321171/3032, we have equidistribution of the two parameters $$a(G) = \begin{cases} 1 & \text{if G has no vertices of degree 1}\\ 0 & \text{otherwise} \end{cases}$$
$$b(G) = \begin{cases} 1 & \text{if G has no two vertices with the same set of neighbours}\\ 0 & \text{otherwise} \end{cases}$$