The number of dominating sets of vertices of a graph. This is, the number of subsets of vertices such that every vertex is either in this subset or adjacent to an element therein [1].

Half the number of sets of vertices in a graph which are dominating and non-blocking. A set of vertices $U$ in a graph is dominating, if every vertex not in $U$ is adjacent to a vertex in $U$. A set of vertices $U$ in a graph is non-blocking, if every vertex in $U$ is adjacent to a vertex not in $U$. Therefore, a set of vertices is non-blocking if and only if its complement is dominating. In particular, if a set of vertices is dominating and non-blocking, so is its complement.

The energy of a graph, if it is integral. The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3]. The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.

The number of distinct eigenvalues of the distance Laplacian of a connected graph.

The pebbling number of a connected graph.

The diameter of a connected graph. This is the greatest distance between any pair of vertices.

The radius of a connected graph. This is the minimum eccentricity of any vertex.

The determinant of the distance matrix of a connected graph.

The hyper-Wiener index of a connected graph. This is $$ \sum_{\{u,v\}\subseteq V} d(u,v)+d(u,v)^2. $$

The largest multiplicity of a 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 $2$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.