The number of acyclic orientations of a graph.

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.

The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0).

The absolute value of the derivative of the chromatic polynomial of the graph at 1. This is closely related to Crapo's beta invariant, the only difference being the value for the graphs without edges.

The beta invariant of the graph. The beta invariant was introduced by Crapo [1] for matroids. For graphs with $n$ vertices the beta invariant is $$\beta(G) = (-1)^{n-c}\sum_{S\subseteq E} (-1)^{|S|} (n-c(S)),$$ where $c(S)$ is the number of connected components of the subgraph of $G$ with edge set $S$. For graphs with at least one edge the beta invariant equals the absolute value of the derivative of the chromatic polynomial at $1$. [2] The beta invariant also coincides with the coefficient of the monomial $x$, and also with the coefficient of the monomial $y$, of the Tutte polynomial.

The number of cover relations in a poset. Equivalently, this is also the number of edges in the Hasse diagram [1].

The pebbling number of a connected graph.

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 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.