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

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.