The number of Hasse diagrams with a given underlying undirected graph. In particular, this statistic vanishes if the graph contains a triangle. This is the size of the preimage of [[Mp00074]].

The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.

The number of Hamiltonian cycles in a graph. A Hamiltonian cycle in a graph $G$ is a subgraph (this is, a subset of the edges) that is a cycle which contains every vertex of $G$. Since it is unclear whether the graph on one vertex is Hamiltonian, the statistic is undefined for this graph.

The toughness times the least common multiple of 1,...,n-1 of a non-complete graph. A graph $G$ is $t$-tough if $G$ cannot be split into $k$ different connected components by the removal of fewer than $tk$ vertices for all integers $k>1$. The toughness of $G$ is the maximal number $t$ such that $G$ is $t$-tough. It is a rational number except for the complete graph, where it is infinity. The toughness of a disconnected graph is zero. This statistic is the toughness multiplied by the least common multiple of $1,\dots,n-1$, where $n$ is the number of vertices of $G$.

The normalized isoperimetric number of a graph. The isoperimetric number, or Cheeger constant, of a graph $G$ is $$ i(G) = \min\left\{\frac{|\partial A|}{|A|}\ : \ A\subseteq V(G), 0 < |A|\leq |V(G)|/2\right\}, $$ where $$ \partial A := \{(x, y)\in E(G)\ : \ x\in A, y\in V(G)\setminus A \}. $$ This statistic is $i(G)\cdot\lfloor n/2\rfloor$.