Trivalent expanders, $(\Delta-Y)$-transformation, and hyperbolic surfaces

TL;DR

This paper introduces a new family of trivalent expander graphs derived from Cayley graphs of nilpotent groups using $(\Delta-Y)$-transformations, which tessellate hyperbolic surfaces and exhibit unique combinatorial, topological, and spectral properties.

Contribution

The authors construct novel trivalent expanders tessellating hyperbolic surfaces, expanding the understanding of graph transformations and their geometric implications.

Findings

Graphs tessellate hyperbolic surfaces with large isometry groups

Comparison shows these graphs differ significantly from Platonic graphs

Spectral and topological properties are thoroughly analyzed

Abstract

We construct a new family of trivalent expanders tessellating hyperbolic surfaces with large isometry groups. These graphs are obtained from a family of Cayley graphs of nilpotent groups via $(\Delta-Y)$-transformations. We compare this family with Platonic graphs and their associated hyperbolic surfaces and see that they are generally very different with only one hyperbolic surface in the intersection. Moreover, we study combinatorial, topological and spectral properties of our trivalent graphs and their associated hyperbolic surfaces.