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 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].

The number of non-empty connected induced subgraphs of a graph. More precisely, this is the number of non-empty subsets of the set of vertices of a graph, such that the induced subgraph is connected.

The number of non-isomorphic minors of a graph. A minor of a graph $G$ is a graph obtained from $G$ by repeatedly deleting or contracting edges, or removing isolated vertices. This statistic records the total number of (non-empty) non-isomorphic minors of a graph.

The Wiener index of a graph. This is the sum of the distances of all pairs of vertices.

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

The number of connected components of the friends and strangers graph. Let $X$ and $Y$ be graphs with the same vertex set $\{1,\dots,n\}$. Then the friends-and-strangers graph has as vertex set the set of permutations $\mathfrak S_n$ and edges $\left(\sigma, (i, j)\circ\sigma\right)$ if $(i, j)$ is an edge of $X$ and $\left(\sigma(i), \sigma(j)\right)$ is an edge of $Y$. This statistic is the number of connected components of the friends and strangers graphs where $X=Y$. For example, if $X$ is a complete graph the statistic is $1$, if $X$ has no edges, the statistic is $n!$, and if $X$ is the path graph, the statistic is $$\sum_{k=0}^{\lfloor n/2\rfloor} (-1)^k (n-k)!\binom{n-k}{k},$$ see [thm. 2.2, 3].

The number of potential covers of a poset. A potential cover is a pair of uncomparable elements $(x, y)$ which can be added to the poset without adding any other relations. For example, let $P$ be the disjoint union of a single relation $(1, 2)$ with the one element poset $0$. Then the relation $(0, 1)$ cannot be added without adding also $(0, 2)$, however, the relations $(0, 2)$ and $(1, 0)$ are potential covers.

The total number of cycles in a graph. For example, the complete graph on four vertices has four triangles and three different four-cycles.