The maximal cardinality of a set of vertices with the same neighbourhood in a graph. The set of so called mating graphs, for which this statistic equals $1$, is enumerated by [1].

The number of four-cliques in a graph.

The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. A graph is $3$-colourable if and only if in any linear ordering of its vertices, there are no four vertices $a < b < c < d$ such that $(a,b), (b,c)$ and $(c,d)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.

The number of odd automorphisms of a graph. Let $D$ be an arbitrary orientation of a graph $G$. Then an automorphism of $G$ is odd, if it reverses the orientation of an odd number of edges of $D$. The graphs on $n$ vertices without any odd automorphisms are equinumerous with the number of non-isomorphic $n$-team tournaments, see [2]. The odd automorphisms of the complete graphs are precisely the even permutations.

The number of triconnected components of a graph. A connected graph is '''triconnected''' or '''3-vertex connected''' if it cannot be disconnected by removing two or fewer vertices. An arbitrary connected graph can be decomposed as a union of biconnected (2-vertex connected) graphs, known as '''blocks''', and each biconnected graph can be decomposed as a union of components with are either a cycle (type "S"), a cocyle (type "P"), or triconnected (type "R"). The decomposition of a biconnected graph into these components is known as the '''SPQR-tree''' of the graph.