The Galaxies of Nonstandard Enlargements of Infinite and Transfinite Graphs

TL;DR

This paper explores the structure of nonstandard enlargements of infinite and transfinite graphs, revealing conditions under which these enlargements have either one or infinitely many galaxies, and analyzing their ordering properties.

Contribution

It introduces the concept of galaxies in nonstandard enlargements of graphs and characterizes their number and ordering, extending the understanding of infinite graph structures.

Findings

Enlargements have either one galaxy or infinitely many.

Galaxies are partially ordered by their closeness to the principal galaxy.

Existence of totally ordered sequences of galaxies in certain enlargements.

Abstract

The galaxies of nonstandard enlargements of conventionally infinite as well as of transfinite graphs are defined, analyzed, and illustrated by some examples. It is then shown that any such enlargement either has exactly one galaxy, its principal one, or it has infinitely many galaxies. In the latter case, the galaxies are partially ordered by their "closeness" to the principal galaxy. If an enlargement has a galaxy different from its principal galaxy, then it has a two-way infinite sequence of galaxies that are totally ordered according to that "closeness" property. There may be many such totally ordered sequences.