Rad HAZU, Matematičke znanosti, Vol. 24 (2020), 49-58.

SOME PROPERTIES OF THE EXTENDED ZERO-DIVISOR GRAPH OF THE RING OF GAUSSIAN INTEGERS MODULO n

Basem Alkhamaiseh

Department of Mathematics, Yarmouk university, Irbid, Jordan
e-mail: basem.m@yu.edu.jo

Abstract.   Recently, Bennis and others studied an extension of the zero-divisor graph of a commutative ring R. They called this extension the extended zero-divisor graph of R, denoted by Γ(R). The graph Γ(R) has as set of vertices all the nonzero zero-divisors of R, Z(R)*, and two distinct vertices x and y are adjacent if there are nonnegative integers n and m such that xnym = 0 with xn ≠ 0 and ym ≠ 0. In this paper, we study several properties of the extended zero-divisor graph of the ring of Gaussian integers modulo n (Γ(Zn[i])). We characterize the positive integers n such that Γ(Zn[i]) = Γ(Zn[i]). The diameter and girth, as well as the positive integers n such that Γ(Zn[i]) is planar or outerplanar, are also determined.

2020 Mathematics Subject Classification.   13A99, 13B99, 05C25.

Key words and phrases.   Ring of Gaussian integers modulo n, extended zero-divisor graph, diameter and girth, planar graph.

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DOI: https://doi.org/10.21857/9e31lhvggm

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