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 de-duplicate of a graph.
Let $G = (V, E)$ be a graph. This map yields the graph whose vertex set is the set of (distinct) neighbourhoods $\{N_v | v \in V\}$ of $G$, and has an edge $(N_a, N_b)$ between two vertices if and only if $(a, b)$ is an edge of $G$. This is well-defined, because if $N_a = N_c$ and $N_b = N_d$, then $(a, b)\in E$ if and only if $(c, d)\in E$.
The image of this map is the set of so-called 'mating graphs' or 'point-determining graphs'.
This map preserves the chromatic number.

Return an ordered tree of size $n+1$ by the following recursive rule:

• if $x$ is the left child of $y$, $x$ becomes the left brother of $y$,
• if $x$ is the right child of $y$, $x$ becomes the last child of $y$.

Return the undirected graph obtained from the tree nodes and edges.