A note on the zero divisor graph of the ring of Lipschitz integers modulo $n$

TL;DR

This paper completes the analysis of the zero divisor graph of Lipschitz integers modulo prime powers, determining the automorphism group for powers of two, thus extending previous work on the domination number.

Contribution

It solves the open problem of characterizing the zero divisor graph for Lipschitz integers modulo prime powers and identifies the automorphism group for powers of two.

Findings

Determined the zero divisor graph structure for Lipschitz integers modulo prime powers.

Computed the automorphism group of the zero divisor graph for powers of two.

Extended the understanding of algebraic graph properties in Lipschitz integer rings.

Abstract

In a recent paper, Grau et al. (2017) studied the zero divisor graphs of the ring of Lipschitz integers modulo $n$, and computed the domination number of the undirected zero divisor graph of the ring of Lipschitz integers modulo $n$. But the case $n$ is a power of prime numbers remained open. In this note, this problem is solved. We also show the automorphism group of the zero divisor graph of the ring of Lipschitz integers modulo $2^s$.