Natural Unit Interval Graph

A natural unit interval graph on the ordered vertices is a graph specified by weakly increasing integers , ..., with , having edges exactly when . The specified ordering of the vertices is part of this convention. For the graph has no edges, while gives the complete graph. Choosing gives a path graph. Its chromatic quasisymmetric function is symmetric in the color variables, and its elementary expansion is the subject of the Shareshian-Wachs conjecture.

This graph is the incomparability graph of the unit interval order on in which strictly precedes exactly when . Its edges join precisely the pairs that are incomparable in this partial order.