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.
The global defensive [Formula: see text]-alliance is a very well-studied notion in graph theory, it provides a method of classification of graphs based on relations between members of a particular set of vertices. In this paper, we explore this notion in zero-divisor graph of commutative rings. The established results generalize and improve recent work by Muthana and Mamouni who treated a particular case for [Formula: see text] known by the global defensive alliance. Various examples are also provided which illustrate and delimit the scope of the established results.
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