The exponent of the automorphism group of a graph. The exponent of a group is the smallest integer $n$ such that $g^n = 1$ for all elements $g \in G$. Thus, this statistic sends a graph $G$ to the least common multiple of all orbit sizes of vertices of $G$ under automorphisms.

The global dimension of the incidence algebra of the lattice over the rational numbers.

The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.

The Schultz index of a connected graph. This is $$\sum_{\{u,v\}\subseteq V} (d(u)+d(v))d(u,v)$$ where $d(u)$ is the degree of vertex $u$ and $d(u,v)$ is the distance between vertices $u$ and $v$. For trees on $n$ vertices, the Schultz index is related to the Wiener index via $S(T)=4W(T)-n(n-1)$ [2].

The second Elser number of a connected graph. For a connected graph $G$ the $k$-th Elser number is $$ els_k(G) = (-1)^{|V(G)|+1} \sum_N (-1)^{|E(N)|} |V(N)|^k $$ where the sum is over all nuclei of $G$, that is, the connected subgraphs of $G$ whose vertex set is a vertex cover of $G$. It is clear that this number is even. It was shown in [1] that it is non-negative.

The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. The deck of a graph is the multiset of induced subgraphs obtained by deleting a single vertex. The graph reconstruction conjecture states that the deck of a graph with at least three vertices determines the graph. This statistic is only defined for graphs with at least two vertices, because there is only a single graph of the given size otherwise.