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 cutwidth of a graph. This is the minimum possible width of a linear ordering of its vertices, where the width of an ordering $\sigma$ is the maximum, among all the prefixes of $\sigma$, of the number of edges that have exactly one vertex in a prefix.

The bandwidth of a graph. The bandwidth of a graph is the smallest number $k$ such that the vertices of the graph can be ordered as $v_1,\dots,v_n$ with $k \cdot d(v_i,v_j) \geq |i-j|$. We adopt the convention that the singleton graph has bandwidth $0$, consistent with the bandwith of the complete graph on $n$ vertices having bandwidth $n-1$, but in contrast to any path graph on more than one vertex having bandwidth $1$. The bandwidth of a disconnected graph is the maximum of the bandwidths of the connected components.

The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. A graph is a disjoint union of paths if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,c)$ is an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.

The dimension of a graph. The dimension of a graph is the least integer $n$ such that there exists a representation of the graph in the Euclidean space of dimension $n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.

The proper pathwidth of a graph. The proper pathwidth $\operatorname{ppw}(G)$ was introduced in [1] as the minimum width of a proper-path-decomposition. Barioli et al. [2] showed that if $G$ has at least one edge, then $\operatorname{ppw}(G)$ is the minimum $k$ for which $G$ is a minor of the Cartesian product $K_k \square P$ of a complete graph on $k$ vertices with a path; and further that $\operatorname{ppw}(G)$ is the minor monotone floor $\lfloor \operatorname{Z} \rfloor(G) := \min\{\operatorname{Z}(H) \mid G \preceq H\}$ of the [[St000482|zero forcing number]] $\operatorname{Z}(G)$. It can be shown [3, Corollary 9.130] that only the spanning supergraphs need to be considered for $H$ in this definition, i.e. $\lfloor \operatorname{Z} \rfloor(G) = \min\{\operatorname{Z}(H) \mid G \le H,\; V(H) = V(G)\}$. The minimum degree $\delta$, treewidth $\operatorname{tw}$, and pathwidth $\operatorname{pw}$ satisfy $$\delta \le \operatorname{tw} \le \operatorname{pw} \le \operatorname{ppw} = \lfloor \operatorname{Z} \rfloor \le \operatorname{pw} + 1.$$ Note that [4] uses a different notion of proper pathwidth, which is equal to bandwidth.

The 2-dynamic chromatic number of a graph. A $k$-dynamic coloring of a graph $G$ is a proper coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-dynamic chromatic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring. This statistic records the $2$-dynamic chromatic number of a graph.

The 3-weak dynamic number of a graph. A $k$-weak-dynamic coloring of a graph $G$ is a (non-proper) coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-weak-dynamic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring. This statistic records the $3$-weak-dynamic number of a graph.

The rank of the adjacency matrix of a graph.

The minimal number of edges to add or remove to make a graph a line graph.