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 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 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.