On zero-divisor graph of the ring of Gaussian integers modulo $2^n$
Aruna Venkatesan, Krishnan Paramasivam, M. Sabeel K
TL;DR
The paper addresses the structure of the zero-divisor graph of the ring $\\mathbb{Z}_{2^n}[i]$ by partitioning zero-divisors into associate classes $V_d$ corresponding to divisors of $2^n$. It adopts a compressed graph viewpoint, revealing a central clique $\\Lambda=\\cup_{j=1}^n V_{d_j}$ and outer blocks $\\Omega=\\cup_{j=n+1}^{2n-1} V_{d_j}$, and derives exact invariants including order $|V|=2^{2n-1}-1$, clique number $2^n-1$, radius $1$, diameter $2$, and chromatic number $\\chi(\\Gamma)=2^n-1$, along with a complete description of edges and planarity conditions. The work further provides explicit maximum and saturation matchings, yielding the saturation number $s(\\Gamma)=2^{n-1}-1$, and computes topological indices such as the Wiener index $W(\\Gamma)$, Randic index $Rand(\\Gamma)$, and Zagreb indices $M_1(\\Gamma)$ and $M_2(\\Gamma)$ via a distance-matrix approach on a compressed graph $\\Gamma_E(\\mathbb{Z}_{2^n}[i])$. Overall, the paper offers a thorough, algebraically structured characterization of the zero-divisor graph for $\\mathbb{Z}_{2^n}[i]$, introducing the associate-class framework as a practical tool for graph-invariant analyses in similar rings.
Abstract
For a commutative ring $R$, the zero-divisor graph of $R$ is a simple graph with the vertex set as the set of all zero-divisors of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy = 0$. This article attempts to predict the structure of the zero-divisor graph of the ring of Gaussian integers modulo $2$ to the power $n$ and determine the size, chromatic number, clique number, independence number, and matching through associate classes of divisors of $2^n$ in $\mathbb{Z}_{2^n}[i]$. In addition, a few topological indices of the corresponding zero-divisor graph, are obtained.