The distant graph of the projective line over a finite ring with unity

TL;DR

This paper investigates the structure of the distant graph formed by the projective line over finite rings with unity, classifying all such graphs for rings up to order p^5, revealing their isomorphism classes.

Contribution

It provides a complete classification of the distant graphs over finite rings of small order, extending understanding of projective lines over rings.

Findings

Classified all distant graphs for rings up to order p^5

Identified isomorphism classes of these graphs

Enhanced understanding of geometric structures over finite rings

Abstract

We discuss the projective line $\mathbb{P}(R)$ over a finite associative ring with unity. $\mathbb{P}(R)$ is naturally endowed with the symmetric and anti-reflexive relation "distant". We study the graph of this relation on $\mathbb{P}(R)$ and classify up to isomorphism all distant graphs $G(R, \Delta)$ for rings $R$ up to order $p^5$, $p$ prime.