The number of ways to place as many non-attacking rooks as possible on a Ferrers board.

The leading coefficient of the rook polynomial of an integer partition. Let $m$ be the minimum of the number of parts and the size of the first part of an integer partition $\lambda$. Then this statistic yields the number of ways to place $m$ non-attacking rooks on the Ferrers board of $\lambda$.

The energy of a graph, if it is integral. The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3]. The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.

The minimal number of edges to add or remove to make a graph regular.

The minimal number of edges to add or remove to make a graph vertex transitive. A graph is vertex transitive if for any two edges there is an automorphism that maps one vertex to the other.

The villainy of a graph. The villainy of a permutation of a proper coloring $c$ of a graph is the minimal Hamming distance between $c$ and a proper coloring. The villainy of a graph is the maximal villainy of a permutation of a proper coloring.