Polytopes from Subgraph Statistics

TL;DR

This paper investigates polytopes derived from subgraph statistics, introducing semi-algebraic sets called curvy zonotopes, and explores their properties through graph limits, volume calculations, and algebraic descriptions to address applications and conjectures in extremal graph theory.

Contribution

It introduces the concept of curvy zonotopes and studies their properties using graph limits, providing new insights into polytopes from subgraph statistics.

Findings

Introduction of curvy zonotopes as semi-algebraic sets

Explicit large polytopes inscribed in subgraph statistic polytopes

Several conjectures based on volume and algebraic analysis

Abstract

Polytopes from subgraph statistics are important in applications and conjectures and theorems in extremal graph theory can be stated as properties of them. We have studied them with a view towards applications by inscribing large explicit polytopes and semi-algebraic sets when the facet descriptions are intractable. The semi-algebraic sets called curvy zonotopes are introduced and studied using graph limits. From both volume calculations and algebraic descriptions we find several interesting conjectures.