Continuity of Functions on Bare Representation of Graphs under Star Topology

TL;DR

This paper introduces a new star topology on finite graphs, examining continuous maps between their Bare representations and highlighting differences from traditional graph homomorphisms.

Contribution

It presents a novel star topology on graphs and analyzes the properties of continuous maps in this new framework, contrasting with classical graph homomorphisms.

Findings

Not all continuous maps correspond to graph homomorphisms.

The star topology unifies vertices and edges in a single space.

Continuous maps can lose adjacency information.

Abstract

This paper introduces a novel topology, referred to as the star topology, on finite graphs. By treating vertices and edges as points in a unified space, we explore continuous maps between Bare representations of a graph and their properties. The key distinction lies in the fact that while every graph homomorphism induces a continuous map, the converse is not generally true due to potential loss of adjacency information.