TITLE: equidistributed parameters on graphs
QUESTION [7 upvotes]: 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.

REPLY [4 votes]: 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.