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Let R be a commutative ring with zero-divisors [Formula: see text] and [Formula: see text] be a positive integer. The [Formula: see text]-extended zero-divisor graph of [Formula: see text], denoted by [Formula: see text], is the (simple) graph with vertex set [Formula: see text], the set of nonzero zero-divisors of [Formula: see text], and two distinct nonzero zero-divisors [Formula: see text] and [Formula: see text] are adjacent whenever there exist two positive integers [Formula: see text] such that [Formula: see text] with [Formula: see text] and [Formula: see text]. The [Formula: see text]-extended zero-divisor graph of [Formula: see text] is well studied in [10]. In this paper, we characterize the [Formula: see text]-extended zero-divisor graphs of idealizations [Formula: see text] (where [Formula: see text] is an [Formula: see text]-module). Namely, we study in detail the behavior of the filtration [Formula: see text] as well as the relations between its terms. We also characterize the girth and the diameter of [Formula: see text] and we give answers to several interesting and natural questions that arise in this context.
Let R be a commutative ring with zero-divisors [Formula: see text] and [Formula: see text] be a positive integer. The [Formula: see text]-extended zero-divisor graph of [Formula: see text], denoted by [Formula: see text], is the (simple) graph with vertex set [Formula: see text], the set of nonzero zero-divisors of [Formula: see text], and two distinct nonzero zero-divisors [Formula: see text] and [Formula: see text] are adjacent whenever there exist two positive integers [Formula: see text] such that [Formula: see text] with [Formula: see text] and [Formula: see text]. The [Formula: see text]-extended zero-divisor graph of [Formula: see text] is well studied in [10]. In this paper, we characterize the [Formula: see text]-extended zero-divisor graphs of idealizations [Formula: see text] (where [Formula: see text] is an [Formula: see text]-module). Namely, we study in detail the behavior of the filtration [Formula: see text] as well as the relations between its terms. We also characterize the girth and the diameter of [Formula: see text] and we give answers to several interesting and natural questions that arise in this context.
Let [Formula: see text] be a commutative ring with nonzero identity and [Formula: see text] be the set of all nonzero zero-divisors of [Formula: see text]. The zero-divisor graph of [Formula: see text], denoted by [Formula: see text], is a simple graph whose vertex set consists of all elements of [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we investigate the [Formula: see text]-spectral of the zero-divisor graph of the ring [Formula: see text]. Specially, we study the [Formula: see text]-spectral of the zero-divisor graph of the ring [Formula: see text] for [Formula: see text], [Formula: see text], [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text] are distinct prime numbers and [Formula: see text]. Moreover, we study the [Formula: see text]-spectral of the zero-divisor graph of the ring [Formula: see text], where [Formula: see text] is a prime number and [Formula: see text].
In 2008, J. Skowronek-kazi o ´ w extended the study of the clique number ω G Z n to the zero-divisor graph of the ring Z n , but their result was imperfect. In this paper, we reconsider ω G Z n of the ring Z n and give some counterexamples. We propose a constructive method for calculating ω G Z n and give an algorithm for calculating the clique number of zero-divisor graph. Furthermore, we consider the case of the ternary zero-divisor and give the generation algorithm of the ternary zero-divisor graphs.
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