ISOPERIMETRIC CONSTANTS IN PLANAR GRAPHS WITH HYPERBOLIC PROPERTIES

Abstract

Isoperimetric inequalities date back to ancient Greece where figures with equal perimeters but different shapes were compared (Zenodorus, On Isoperimetric Figures). The original problem was to maximize the area contained within a curve of specified length. In Euclidean geometry the result is a circle. This can be generalized to shapes on non-Euclidean surfaces as well as for higher dimensions where we seek to maximize the hyperdimensional volume respective to the hyperdimensional surface area.The subject of this thesis is isoperimetric constants in planar graphs with hyperbolic properties. We first analyze isoperimetric constants in the flat plane which has curvature 0 and then give an overview of two isoperimetric constants that give a hyperbolicity criterion for infinite vertex-regular, face-regular planar graphs. Finally we extend the concept to general planar graphs with hyperbolic properties and establish that the same constants serve as a lower bound.