The number of acyclic orientations of a graph.

The number of forests contained in a graph. That is, for a graph $G = (V,E)$ with vertices $V$ and edges $E$, the number of subsets $E' \subseteq E$ for which the subgraph $(V,E')$ is acyclic. If $T_G(x,y)$ is the Tutte polynomial [2] of $G$, then the number of forests contained in $G$ is given by $T_G(2,1)$.

The composition number of a graph. This is the number of set partitions of the vertex set of the graph, such that the subgraph induced by each block is connected.

The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1).

The number of binary trees with the same underlying unordered tree.

The pebbling number of a connected graph.

The number of spanning subgraphs of a graph. This is the number of subsets of the edge set of the graph, or the evaluation of the Tutte polynomial at $x=y=2$.

The chromatic discriminant of a graph. The chromatic discriminant $\alpha(G)$ is the coefficient of the linear term of the chromatic polynomial $\chi(G,q)$. According to [1], it equals the cardinality of any of the following sets: (1) Acyclic orientations of G with unique sink at $q$, (2) Maximum $G$-parking functions relative to $q$, (3) Minimal $q$-critical states, (4) Spanning trees of G without broken circuits, (5) Conjugacy classes of Coxeter elements in the Coxeter group associated to $G$, (6) Multilinear Lyndon heaps on $G$. In addition, $\alpha(G)$ is also equal to the the dimension of the root space corresponding to the sum of all simple roots in the Kac-Moody Lie algebra associated to the graph.