The number of edges of a graph.

The Szeged index of a graph.

The Wiener index of a graph. This is the sum of the distances of all pairs of vertices.

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 major index east count of a Dyck path. The descent set $\operatorname{des}(D)$ of a Dyck path $D = D_1 \cdots D_{2n}$ with $D_i \in \{N,E\}$ is given by all indices $i$ such that $D_i = E$ and $D_{i+1} = N$. This is, the positions of the valleys of $D$. The '''major index''' of a Dyck path is then the sum of the positions of the valleys, $\sum_{i \in \operatorname{des}(D)} i$, see [[St000027]]. The '''major index east count''' is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = E\}$.

The sum of the positions of double up-steps of a Dyck path. This is part of MacMahon's equal index of a word, see [1, p. 135].

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 number of maximal dissociation sets in a graph.

The difference in the number of possibilities of choosing a pair of negative eigenvalues and the signature of a graph. Let $n^+(G)$, $n^0(G)$ and $n^-(G)$ denote the number of positive, zero, and negative eigenvalues of $G$. The signature of a graph $s(G)$ is $n^+(G) - n^-(G)$. This statistic records $\binom{n^-(G)+1}{2} - n^+(G) = \binom{n^-(G)}{2} - s(G)$. In [1], it is conjectured to be non-negative.

The major index of the composition. The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents. For details about the major index see [[Permutations/Descents-Major]].