The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,...,n). Let $\rho=(1,\dots,n)$ and $\sigma=(1,2)$. Then, for a permutation $\pi\in\mathfrak S_n$, this statistic is $$\min\{k\mid \pi=\rho^{i_0}\sigma\rho^{i_1}\sigma\dots\rho^{i_{k-1}}\sigma\rho^{i_k}, 0\leq i_0,\dots,i_k < n\}.$$ Put differently, it is the minimal length of a factorization into cyclic shifts of the transposition $(1,2)$ (see [[St001076]]) of any of the permutations $\rho^k\pi$ for $0\leq k < n$.

The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.

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 largest multiplicity of a distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $2$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.

The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.

The diameter of a connected graph. This is the greatest distance between any pair of vertices.