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

The competition number of a graph. The competition graph of a digraph $D$ is a (simple undirected) graph which has the same vertex set as $D$ and has an edge between $x$ and $y$ if and only if there exists a vertex $v$ in $D$ such that $(x, v)$ and $(y, v)$ are arcs of $D$. For any graph, $G$ together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number $k(G)$ is the smallest number of such isolated vertices.

The Euler characteristic of a graph. The '''Euler characteristic''' $\chi$ of a topological space is the alternating sum of the dimensions of the homology groups $$\chi(X) = \sum_{k \geq 0} (-1)^k \dim H_k(X).$$ For a finite simplicial complex, this is equal to the alternating sum $\sum_{k\geq 0} (-1)^k f_k$ where $f_k$ the number of $k$-dimensional simplices. A (simple) graph is a simplicial complex of dimension at most one; its vertices are the 0-simplices and its edges are the 1-simplices. For a connected graph, the Euler characteristic is equal to $1 - g$ where $g$ is the cyclomatic number.

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.