The degree of the minimal polynomial of the largest eigenvalue of a graph.

The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. The disjoint direct product decomposition of a permutation group factors the group corresponding to the product $(G, X) \ast (H, Y) = (G\times H, Z)$, where $Z$ is the disjoint union of $X$ and $Y$. In particular, for an asymmetric graph, i.e., with trivial automorphism group, this statistic equals the number of vertices, because the trivial action factors completely.

The number of occurrences of the pattern 13-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.

The diameter of a connected graph. This is the greatest distance between any pair of vertices.

The number of shortest chains of small intervals from the bottom to the top in a lattice. An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.

The size of the image of the pop stack sorting operator. The pop stack sorting operator is defined by $Pop_L^\downarrow(x) = x\wedge\bigwedge\{y\in L\mid y\lessdot x\}$. This statistic returns the size of $Pop_L^\downarrow(L)\}$.

The minimal length of a chain of small intervals in a lattice. An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.

The number of elements which do not have a complement in the lattice. A complement of an element $x$ in a lattice is an element $y$ such that the meet of $x$ and $y$ is the bottom element and their join is the top element.

The pebbling number of a connected 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.