Flip Graphs

TL;DR

This thesis introduces Yoke graphs, a new family of flip graphs generalizing several known types, and provides their diameter and automorphism group, revealing new structural insights and connections to affine Weyl groups.

Contribution

It introduces Yoke graphs, computes their diameter, characterizes their automorphism group, and links them to affine Weyl groups, extending prior work on flip graphs.

Findings

Yoke graphs' diameters are computed.

Automorphism groups of Yoke graphs are characterized.

Yoke graphs are Schreier graphs of affine Weyl groups.

Abstract

Flip graphs are graphs on combinatorial objects in which the adjacency relation reflects a local change in the underlying objects. In this thesis we introduce Yoke graphs, a family of flip graphs that generalizes previously studied families of flip graphs on colored triangle-free triangulations, arc permutations and geometric caterpillars. Our main results are the computation of the diameter of an arbitrary Yoke graph and a full characterization of the automorphism group of this family of graphs. We also show that Yoke graphs are Schreier graphs of the affine Weyl group of type $\tilde{C}_m$. The approach we take in the computation of the diameter is different from the ones used for colored triangle-free triangulations and arc permutations. We show that the approach used for arc permutation graphs does not extend to Yoke graphs. At the heart of our proof lies the idea of transforming a diameter evaluation into an eccentricity problem. The characterization of the automorphism group is a new result for the above mentioned three special families of Yoke graphs.