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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.

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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} $

and

$ b(G) = \begin{cases} 1 & \text{if $G$ has no two vertices with the same set of neighbours}\\ 0 & \text{otherwise} \end{cases} $

This was shown bijectively by Kilibarda. His bijection preserves connectedness (but I have not yet understood it).

Kilibarda, Goran, Enumeration of unlabelled mating graphs, Graphs Comb. 23, No. 2, 183-199 (2007). ZBL1116.05038.

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