The sum of the vertex degrees of a graph. This is clearly equal to twice the number of edges, and, incidentally, also equal to the trace of the Laplacian matrix of a graph. From this it follows that it is also the sum of the squares of the eigenvalues of the adjacency matrix of the graph. The Laplacian matrix is defined as $D-A$ where $D$ is the degree matrix (the diagonal matrix with the vertex degrees on the diagonal) and where $A$ is the adjacency matrix. See [1] for detailed definitions.

The major index of a Dyck path. This is the sum over all indices $i$, starting with 1, such that the $i$-th step is a down step and the $(i+1)$-st step is an up step. The generating function of the major index yields '''MacMahon''' 's $q$-Catalan numbers $$\sum_{D \in \mathfrak{D}_n} q^{\operatorname{maj}(D)} = \frac{1}{[n+1]_q}\begin{bmatrix} 2n \\ n \end{bmatrix}_q,$$ where $[k]_q := 1+q+\ldots+q^{k-1}$ is the usual $q$-extension of the integer $k$, $[k]_q!:= [1]_q[2]_q \cdots [k]_q$ is the $q$-factorial of $k$ and $\left[\begin{smallmatrix} k \\ l \end{smallmatrix}\right]_q:=[k]_q!/[l]_q![k-l]_q!$ is the $q$-binomial coefficient. The major index was first studied by P.A.MacMahon in [1], where he proved this generating function identity. There is a bijection $\psi$ between Dyck paths and '''noncrossing permutations''' which simultaneously sends the area of a Dyck path [[St000012]] to the number of inversions [[St000018]], and the major index of the Dyck path to $n(n-1)$ minus the sum of the major index and the major index of the inverse [2]. For the major index on other collections, see [[St000004]] for permutations and [[St000290]] for binary words.

Half of MacMahon's equal index of a Dyck path. This is half the sum of the positions of double (up- or down-)steps of a Dyck path, see [1, p. 135].

The sum of the major and the inverse major index of a permutation. This statistic is the sum of [[St000004]] and [[St000305]].

The natural major index of a standard Young tableau. A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation. The natural major index of a tableau with natural descent set $D$ is then $\sum_{d\in D} d$.

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 number of B-inversions of a signed permutation. The number of B-inversions of a signed permutation $\sigma$ of length $n$ is $$ \operatorname{inv}_B(\sigma) = \big|\{ 1 \leq i < j \leq n \mid \sigma(i) > \sigma(j) \}\big| + \big|\{ 1 \leq i \leq j \leq n \mid \sigma(-i) > \sigma(j) \}\big|, $$ see [1, Eq. (8.2)]. According to [1, Eq. (8.4)], this is the Coxeter length of $\sigma$.

The toughness times the least common multiple of 1,...,n-1 of a non-complete graph. A graph $G$ is $t$-tough if $G$ cannot be split into $k$ different connected components by the removal of fewer than $tk$ vertices for all integers $k>1$. The toughness of $G$ is the maximal number $t$ such that $G$ is $t$-tough. It is a rational number except for the complete graph, where it is infinity. The toughness of a disconnected graph is zero. This statistic is the toughness multiplied by the least common multiple of $1,\dots,n-1$, where $n$ is the number of vertices of $G$.