Flag graph

A flag graph is a way to express the structure of a polytopes flags.

The flag graph of an abstract polytope is a maniplex, although not all mainplexes are the flag graph of a polytope.

Definition
Two flags of a polytope are $j$-adjacent if they differ in exactly one element of rank $j$.

The flag graph is a edge colored graph with vertices corresponding to the flags of a polytope and with $i$ colored edges between flags that are $i$-adjacent.

Properties

 * A polytope is dyadic iff every vertex in its flag graph has exactly one edge of every color.
 * Let $P$ be the flag graph of an abstract polytope, for $i$ and $j$ which differ by more than one, the subgraph of $P$ comprised of edges which are colored $i$ or $j$ consists entirely of disconnected cycles of length 4.