The game chromatic number of a graph. Two players, Alice and Bob, take turns colouring properly any uncolored vertex of the graph. Alice begins. If it is not possible for either player to colour a vertex, then Bob wins. If the graph is completely colored, Alice wins. The game chromatic number is the smallest number of colours such that Alice has a winning strategy.

The radius of a connected graph. This is the minimum eccentricity of any vertex.

The number of vertices of the largest induced star graph in the graph.

The number of proper colouring schemes of a Ferrers diagram. A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic is the number of distinct such integer partitions that occur.

The multiplicity of the largest part of an integer partition.

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

The number of triangles of a graph. A triangle $T$ of a graph $G$ is a collection of three vertices $\{u,v,w\} \in G$ such that they form $K_3$, the complete graph on three vertices.

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 minimal crossing number of a graph. A '''drawing''' of a graph $G$ is a drawing in $\mathbb{R}^2$ such that * the vertices of $G$ are distinct points, * the edges of $G$ are simple curves joining their endpoints, * no edge passes through a vertex, and * no three edges cross in a common point. The '''minimal crossing number''' of $G$ is then the minimal number of crossings of edges in a drawing of $G$. In particular, a graph is planar if and only if its minimal crossing number is $0$. It is moreover conjectured that the crossing number of the complete graph $K_n$ [1] is $$\frac{1}{4}\lfloor \frac{n}{2} \rfloor\lfloor \frac{n-1}{2} \rfloor\lfloor \frac{n-2}{2} \rfloor\lfloor \frac{n-3}{2} \rfloor,$$ and the crossing number of the complete bipartite graph $K_{n,m}$ [2] is $$\lfloor \frac{n}{2} \rfloor\lfloor \frac{n-1}{2} \rfloor\lfloor \frac{m}{2} \rfloor\lfloor \frac{m-1}{2} \rfloor.$$ A general algorithm to compute the crossing number is e.g. given in [3]. This statistics data was provided by Markus Chimani [6].