The maximin edge-connectivity for choosing a subgraph. This is $\max_X \min(\lambda(G[X]), \lambda(G[V\setminus X]))$, where $X$ ranges over all subsets of the vertex set $V$ and $\lambda$ is the edge-connectivity of a graph.

The number of ways to refine the partition into singletons. For example there is only one way to refine $[2,2]$: $[2,2] > [2,1,1] > [1,1,1,1]$. However, there are two ways to refine $[3,2]$: $[3,2] > [2,2,1] > [2,1,1,1] > [1,1,1,1,1$ and $[3,2] > [3,1,1] > [2,1,1,1] > [1,1,1,1,1]$. In other words, this is the number of saturated chains in the refinement order from the bottom element to the given partition. The sequence of values on the partitions with only one part is [[A002846]].

The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1).

The number of graphs with the given composition of multiplicities of Laplacian eigenvalues.

The cardinality of a maximal independent set of vertices of a graph. An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.

The side length of the Durfee square of an integer partition. Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$. This is also known as the Frobenius rank.

The number of orbits of vertices of a graph under automorphisms.

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 length of the longest staircase fitting into an integer composition. For a given composition $c_1,\dots,c_n$, this is the maximal number $\ell$ such that there are indices $i_1 < \dots < i_\ell$ with $c_{i_k} \geq k$, see [def.3.1, 1]

The maximal number of occurrences of a colour in a proper colouring 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 largest part occurring in any of these partitions. 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 $[3,2,1]$. Therefore, the statistic on this graph is $3$.