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

Abstract

Abstract. This paper is a continuation for the study of the zero-divisor graph for the ring of Gaussian integers modulo n, ‫ޚ(‬ n [i] Let n be a natural number and let < n > be the principal ideal generated by n in ‫ [ޚ‬i]. Then the factor ring

Some properties of the zero divisor graph of a commutative ring

“…Then, the core of Γ(R) is the maximal clique in Γ(R), [16]. On the other hand, χ(Γ(Z n [i])), and ω(Γ(Z n [i])), when n is a power of a prime are computed in, [3] and [18] respectively. Furthermore, the maximal clique, when n is a power of a prime, is determined in [18] comparing the results in the two papers, we see that, χ(Γ(Z n [i])), and ω(Γ(Z n [i])) are equal, and so we get,…”

“…Proof. (1) From [3], we have Γ(Z 2 m [i]) ∼ = Γ(Z 2 2m ). Thus the result holds by Theorem 31 of [22].…”

“…Abu Osba et al [2] introduced the zero divisor graph for the ring of Gaussian integers modulo n, in which the number of vertices, the diameter, and the girth are found, as well as a complete characterization of the n for which Γ(Z n [i]) is Eulerian or planar. Further properties of Γ(Z n [i]) are investigated in [3]. The complement of Γ(Z n [i]) is studied in [1].…”

“…The eccentricity of each vertex in Γ(Z n [i]), when n is a power of a prime, is determined in [25]. If n = q 1 q 2 , and n is divisible by at least two distinct primes, then, diam(Γ(Z n [i])) = 3, [3], this together with the above theorem give the eccentricity of each vertex in Γ(Z n [i]) when n is divisible by at least two distinct primes.…”

On the Zero Divisor Graphs of the Ring of Lipschitz Integers Modulo n

“…The zero divisor graph of the ring of Gaussian integers modulo n has recently received great attention [1,2,22].…”