Weak Zero-Divisor Graphs of Finite Commutative Rings
DOI:
https://doi.org/10.26713/cma.v15i1.2498Keywords:
Weak zero-divisor graphs, Zero-divisor graphs, Ring of integers modulo n, Weak-nil ring, Nilpotent elementsAbstract
This paper introduces weak zero-divisor graphs of finite commutative rings. The concept of zero-divisor graphs over rings has been extensively investigated for decades. These graphs are constructed with the zero-divisors of rings as their vertex set. We define a weak zero-divisor graph as a graph whose vertex set consists of nonzero elements \(u\) and \(v\) of a ring \(R\) and such that the vertices \(u\) and \(v\) are adjacent if and only if \((uv)^n=0\) for some positive integer \(n\). This article will study parameters of weak zero-divisor graphs of commutative rings, including their diameter, their girth, their radius, their center, and their domination number. We also determine whether weak zero-divisor graphs of the ring of integers \(\mathbb Z_m\) are Eulerian and Hamiltonian.
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