Unit distance graphs and algebraic integers

TL;DR

This paper investigates the structure of unit distance graphs formed by finitely generated additive subgroups of the plane, providing answers to a specific question about vertex degrees posed by Brass.

Contribution

It offers new insights into the properties of unit distance graphs related to algebraic integers and finitely generated groups, addressing an open question in the field.

Findings

Resolved Brass's question on vertex degrees

Characterized unit distance graphs of finitely generated subgroups

Connected algebraic integers to geometric graph properties

Abstract

We answer a question of Brass about vertex degrees in unit distance graphs of finitely generated additive subgroups of $\mathbb{R}^2$.