The number of edges minus the number of vertices plus 2 of a graph. When G is connected and planar, this is also the number of its faces. When $G=(V,E)$ is a connected graph, this is its $k$-monochromatic index for $k>2$: for $2\leq k\leq |V|$, the $k$-monochromatic index of $G$ is the maximum number of edge colors allowed such that for each set $S$ of $k$ vertices, there exists a monochromatic tree in $G$ which contains all vertices from $S$. It is shown in [1] that for $k>2$, this is given by this statistic.

The cyclomatic number of a graph. This is the minimum number of edges that must be removed from the graph so that the result is a forest. This is also the first Betti number of the graph. It can be computed as $c + m - n$, where $c$ is the number of connected components, $m$ is the number of edges and $n$ is the number of vertices.

The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.

The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.

The pebbling number of a connected graph.

The number of cyclic valleys and cyclic peaks of a permutation. This is given by the number of indices $i$ such that $\pi_{i-1} > \pi_i < \pi_{i+1}$ with indices considered cyclically. Equivalently, this is the number of indices $i$ such that $\pi_{i-1} < \pi_i > \pi_{i+1}$ with indices considered cyclically.

The number of distinct colouring schemes of a graph. To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the number of distinct partitions that occur. For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$.

The number of graphs with the same chromatic polynomial.

The cardinality of a minimal edge-isolating set of a graph. Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$. This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains only the graph with one edge.

The number of graphs with the same symmetric edge polytope as the given graph. The symmetric edge polytope of a graph on $n$ vertices is the polytope in $\mathbb R^n$ defined as the convex hull of $e_i - e_j$ and $e_j - e_i$ for each edge $(i, j)$, where $e_1,\dots, e_n$ denotes the standard basis.