Zero-divisor ideals and realizable zero-divisor graphs

Axtell, Michael; Stickles, Joe; Trampbachls, Wallace

doi:10.2140/involve.2009.2.17

Cited by 19 publications

(5 citation statements)

References 3 publications

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“….3.5);x)‫|‬ x=1=351+504x+39705x 2 +21216x 3 . By definition Wiener index of zero divisor graph of Г(Z2 5 .3.5) . W(Г(Z2 5 .3.5)) =504+2(39705)+3(21216) =143562.…”

Section: Proofmentioning

confidence: 99%

Zero Divisor Graph Of ZpM qr with Applications

Shuker¹,

Mohammad²,

Khaleel³

2020

AL-Rafidain Journal of Computer Sciences and Mathematics

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“….3.5);x)‫|‬ x=1=351+504x+39705x 2 +21216x 3 . By definition Wiener index of zero divisor graph of Г(Z2 5 .3.5) . W(Г(Z2 5 .3.5)) =504+2(39705)+3(21216) =143562.…”

Section: Proofmentioning

confidence: 99%

Zero Divisor Graph Of ZpM qr with Applications

Shuker¹,

Mohammad²,

Khaleel³

2020

AL-Rafidain Journal of Computer Sciences and Mathematics

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“…. Cut-sets in the zero-divisor graph of a commutative ring were introduced in [14] and further studied in [8,10,20]. Suppose that R is a commutative ring and that A is a cut-set separating Γ(R) into components X and Y .…”

Section: Cut-sets In γ(R)mentioning

confidence: 99%

Cut structures in zero-divisor graphs of commutative rings

Axtell¹,

Baeth²,

Stickles³

2016

J. Commut. Algebra

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Zero-divisor graphs, and more recently, compressed zero-divisor graphs are well represented in the commutative ring literature. In this work, we consider various cut structures, sets of edges or vertices whose removal disconnects the graph, in both compressed and noncompressed zero-divisor graphs. In doing so, we connect these graph-theoretic concepts with algebraic notions and provide realization theorems of zero-divisor graphs for commutative rings with identity. 2010 AMS Mathematics subject classification. Primary 13A99.

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“…In a connected graph, a cut-vertex is a vertex that, when it and any edges incident to it are removed, separates the graph into two or more connected components. Cut-vertices were introduced into the analysis of zero-divisor graphs in [Axtell et al 2009] and were further studied in [Axtell et al 2011]. In [Redmond 2003, Theorem 3.2], Redmond proved that I (R) contains no cut-vertices whenever I is a nonzero proper ideal of R. Cut-sets, a generalization of the cut-vertex, were also introduced into the analysis of zero-divisor graphs in [Coté et al 2011].…”

Section: Cut-sets and Connectivitymentioning

confidence: 99%

“…Using Theorem 1.1 and Corollary 1.2, there exists an element x + I that is connected to every other element, including itself, in (R/I ). Then, applying Lemma 3.1 in [Axtell et al 2009] gives us that Z (R/I ) is an ideal. If the zero-divisors of a finite ring form an ideal, then that ideal is the maximal ideal of the ring, and the ring is local.…”

mentioning

confidence: 99%

An exploration of ideal-divisor graphs

Axtell

Stickles

Bloome

et al. 2015

Involve

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Zero-divisor graphs have given some interesting insights into the behavior of commutative rings. Redmond introduced a generalization of the zero-divisor graph called an ideal-divisor graph. This paper expands on Redmond's findings in an attempt to find additional information about the structure of commutative rings from ideal-divisor graphs. Definitions and introductionThroughout, we assume that R is a finite commutative ring with identity, though in some instances the proofs given can be extended to more general rings. A zerodivisor in R is an element x such that there exists a nonzero y ∈ R with x y = 0. The set of all zero-divisors in R is denoted by Z (R). The set of all nonzero zero-divisors is denoted by Z (R) * .A graph G is defined by a vertex set V (G) and an edge setTwo vertices x and y joined by an edge are said to be adjacent, denoted x − y. A vertex x is said to be looped if x − x. A path between two elements a 1 , a n ∈ V (G) is an ordered sequence {a 1 , a 2 , . . . , a n } of distinct vertices of G such that a i−1 − a i for all 1 < i ≤ n. If there exists a path between any two distinct vertices, then the graph is said to be connected. A graph is said to be complete if every vertex is adjacent to every other vertex, and we denote the complete graph on n vertices byIf the vertices of a graph G can be partitioned into two sets with vertices adjacent only if they are in distinct sets, then G is bipartite. If vertices in a bipartite graph are adjacent if and only if they are in distinct vertex sets, then the graph is called complete bipartite. We will denote the complete bipartite graph with distinct vertex sets of cardinalities m and n by K m,n . A star graph is a complete bipartite graph MSC2010: 13M05.

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Zero-divisor ideals and realizable zero-divisor graphs

Cited by 19 publications

References 3 publications

Zero Divisor Graph Of ZpM qr with Applications

Zero Divisor Graph Of ZpM qr with Applications

Cut structures in zero-divisor graphs of commutative rings

An exploration of ideal-divisor graphs

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