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 $i(G)\cdot\lfloor n/2\rfloor$.

Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.

Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex. See also [[St000205]]. Each value in this statistic is greater than or equal to corresponding value in [[St000205]].

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 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.

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 $G$.