Diameter and Girth of Zero Divisor Graph of Multiplicative Lattices

TL;DR

This paper investigates the properties of the zero divisor graph of multiplicative lattices, establishing conditions for cycle presence and girth, and characterizing the diameter, thereby resolving a conjecture related to reduced rings with multiple minimal primes.

Contribution

It proves that the zero divisor graph of certain multiplicative lattices has girth 3 and characterizes its diameter, settling a conjecture for reduced rings with multiple minimal primes.

Findings

Zero divisor graph contains a cycle under certain conditions.

Girth of the graph is exactly 3 for specified lattices.

Diameter of the graph is characterized.

Abstract

In this paper, we study the zero divisor graph $\Gamma^m(L)$ of a multiplicative lattice L. We prove under certain conditions that for a reduced multiplicative lattice L having more than two minimal prime elements, $\Gamma^m(L)$ contains a cycle and $gr(\Gamma^m(L)) = 3$. This essentially proves that for a reduced ring R with more than two minimal primes, $gr(\mathbb{AG}(R))) = 3$ which settles the conjecture of Behboodi and Rakeei [9]. Further, we have characterized the diameter of $\Gamma^m(L)$.