The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].

The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.

The Ore closure of a graph.
The Ore closure of a connected graph $G$ has the same vertices as $G$, and the smallest set of edges containing the edges of $G$ such that for any two vertices $u$ and $v$ whose sum of degrees is at least the number of vertices, then $(u,v)$ is also an edge.
For disconnected graphs, we compute the closure separately for each component.

The threshold graph corresponding to the composition.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.