The number of vertices in the center of a graph. The center of a graph is the set of vertices whose maximal distance to any other vertex is minimal. In particular, if the graph is disconnected, all vertices are in the certer.

The number of vertices of maximal degree in a graph.

The number of isolated vertices of a graph.

The absolute value of the derivative of the chromatic polynomial of the graph at 1. This is closely related to Crapo's beta invariant, the only difference being the value for the graphs without edges.

The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$: $$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$ This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^\lambda$.

The number of edges in the center of a graph. The center of a graph is the set of vertices whose maximal distance to any other vertex is minimal. In particular, if the graph is disconnected, all vertices are in the certer.

The Euler characteristic of a graph according to Knill. This is $$\sum_{k\geq 1} (-1)^{k-1} c_k,$$ where $c_k$ is the number of cliques with $k$ vertices.

The number of singletons of an integer partition. A singleton in an integer partition is a part that appear precisely once.

The number of kings in a graph. A vertex of a graph is a king, if all its neighbours have smaller degree. In particular, an isolated vertex is a king.

The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$: $$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$ This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{21^{n-2}}$, for $\lambda\vdash n$.