The spearman's rho of a permutation and the identity permutation. This is, for a permutation $\pi$ of $n$, given by $\sum_{i=1}^n (\pi(i)−i)^2$.

The number of indices that are not cyclical small weak excedances. A cyclical small weak excedance is an index $i < n$ such that $\pi_i = i+1$, or the index $i = n$ if $\pi_n = 1$.

The number of vertices of odd degree in a graph.

The number of leaves in a graph. That is, the number of vertices of a graph that have degree 1.

The sum of the vertex degrees of a graph. This is clearly equal to twice the number of edges, and, incidentally, also equal to the trace of the Laplacian matrix of a graph. From this it follows that it is also the sum of the squares of the eigenvalues of the adjacency matrix of the graph. The Laplacian matrix is defined as $D-A$ where $D$ is the degree matrix (the diagonal matrix with the vertex degrees on the diagonal) and where $A$ is the adjacency matrix. See [1] for detailed definitions.

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 first Zagreb index of a graph. This is the sum of the squares of the degrees of the vertices, $$\sum_{v \in V(G)} d^2(v) = \sum_{\{u,v\}\in E(G)} \big(d(u)+d(v)\big)$$ where $d(u)$ is the degree of the vertex $u$.

The F-index (or forgotten topological index) of a graph. This is $$\sum_{v \in V(G)} d^3(v) = \sum_{\{u,v\}\in E(G)} \big(d^2(u)+d^2(v)\big)$$ where $d(u)$ is the degree of the vertex $u$.

Sum of the even parts of a partition.

The sum of the parts of an integer partition that are at least two.