The girth of a graph, which is not a tree. This is the length of the shortest cycle in the 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 oddness of a signed permutation. The direct sum of two signed permutations $\sigma\in\mathfrak H_k$ and $\tau\in\mathfrak H_m$ is the signed permutation in $\mathfrak H_{k+m}$ obtained by concatenating $\sigma$ with the result of increasing the absolute value of every entry in $\tau$ by $k$. This statistic records the number of blocks with an odd number of signs in the direct sum decomposition of a signed permutation.

The number of Hecke atoms of a signed permutation. For a signed permutation $z\in\mathfrak H_n$, this is the cardinality of the set $$\{ w\in\mathfrak H_n | w^{-1} \star w = z\},$$ where $\star$ denotes the Demazure product. Note that $w\mapsto w^{-1}\star w$ is a surjection onto the set of involutions.

The number of occurrences of the signed pattern 1-2 in a signed permutation. This is the number of pairs $1\leq i < j\leq n$ such that $0 < \pi(i) < -\pi(j)$.

The number of positive entries followed by a negative entry in a signed permutation. For a signed permutation $\pi\in\mathfrak H_n$, this is the number of positive entries followed by a negative entry in $\pi(-n),\dots,\pi(-1),\pi(1),\dots,\pi(n)$.

The reduced word complexity of a permutation. For a permutation $\pi$, this is the smallest length of a word in simple transpositions that contains all reduced expressions of $\pi$. For example, the permutation $[3,2,1] = (12)(23)(12) = (23)(12)(23)$ and the reduced word complexity is $4$ since the smallest words containing those two reduced words as subwords are $(12),(23),(12),(23)$ and also $(23),(12),(23),(12)$. This statistic appears in [1, Question 6.1].