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 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.

Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. The modified algebra B is obtained from the stable Auslander algebra kQ/I by deleting all relations which contain walks of length at least three (conjectural this step of deletion is not necessary as the stable higher Auslander algebras might be quadratic) and taking as B then the algebra kQ^(op)/J when J is the quadratic perp of the ideal I. See http://www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Loewy length and Nakayama algebras associated to Dyck paths.

Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word.

The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. The statistic returns zero in case that bimodule is the zero module.