The second Elser number of a connected graph.
For a connected graph $G$ the $k$-th Elser number is
$$els_k(G) = (-1)^{|V(G)|+1} \sum_N (-1)^{|E(N)|} |V(N)|^k$$
where the sum is over all nuclei of $G$, that is, the connected subgraphs of $G$ whose vertex set is a vertex cover of $G$.
It is clear that this number is even. It was shown in [1] that it is non-negative.

The weak duplicate order of the de-duplicate of a graph.
Let $G=(V, E)$ be a graph and let $N=\{ N_v | v\in V\}$ be the set of (distinct) neighbourhoods of $G$.
This map yields the poset obtained by ordering $N$ by reverse inclusion.

The incomparability graph of a poset.

The complement of a graph.
The complement of a graph has the same vertices, but exactly those edges that are not in the original graph.