The number of pairs of vertices of a graph with distance 4. This is the coefficient of the quartic term of the Wiener polynomial.

The genus of a graph. This is the smallest genus of an oriented surface on which the graph can be embedded without crossings. One can indeed compute the genus as the sum of the genuses for the connected components.

The number of graphs with the same ordinary spectrum as the given graph.

The skewness of a graph. For a graph $G$, the '''skewness''' of $G$ is the minimum number of edges of $G$ whose removal results in a planar graph.

The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. Here $A$ is the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]].

The number of long braid edges in the graph of braid moves of a permutation. Given a permutation $\pi$, let $\operatorname{Red}(\pi)$ denote the set of reduced words for $\pi$ in terms of simple transpositions $s_i = (i,i+1)$. We now say that two reduced words are connected by a long braid move if they are obtained from each other by a modification of the form $s_i s_{i+1} s_i \leftrightarrow s_{i+1} s_i s_{i+1}$ as a consecutive subword of a reduced word. For example, the two reduced words $s_1s_3s_2s_3$ and $s_1s_2s_3s_2$ for $$(124) = (12)(34)(23)(34) = (12)(23)(34)(23)$$ share an edge because they are obtained from each other by interchanging $s_3s_2s_3 \leftrightarrow s_3s_2s_3$. This statistic counts the number of such short braid moves among all reduced words.

The number of connected components of short braid edges in the graph of braid moves of a permutation. Given a permutation $\pi$, let $\operatorname{Red}(\pi)$ denote the set of reduced words for $\pi$ in terms of simple transpositions $s_i = (i,i+1)$. We now say that two reduced words are connected by a short braid move if they are obtained from each other by a modification of the form $s_i s_j \leftrightarrow s_j s_i$ for $|i-j| > 1$ as a consecutive subword of a reduced word. For example, the two reduced words $s_1s_3s_2$ and $s_3s_1s_2$ for $$(1243) = (12)(34)(23) = (34)(12)(23)$$ share an edge because they are obtained from each other by interchanging $s_1s_3 \leftrightarrow s_3s_1$. This statistic counts the number connected components of such short braid moves among all reduced words.