Graphs and Their Vietoris-Rips Complexes Have the Same Pseudotopological Weak Homotopy Type

TL;DR

This paper establishes that graphs and their Vietoris-Rips complexes share the same weak homotopy type, linking graph structures with topological properties of their geometric realizations.

Contribution

It demonstrates a fundamental equivalence between graphs with their ch closure structure and the homotopy type of their Vietoris-Rips complexes.

Findings

Graphs and Vietoris-Rips complexes have the same weak homotopy type.

Establishes a topological equivalence linking graph structures to their geometric realizations.

Provides a theoretical foundation for topological analysis of graphs.

Abstract

In this document, we propose a bridge between the graphs and the geometric realizations of their Vietoris Rips complexes, i.e. Graphs, with their canonical \v{C}ech closure structure, have the same homotopy type that the realization of their Vietoris Rips complex.