On the line graphs of zero-divisor graphs of posets

2016 ◽  
Vol 16 (07) ◽  
pp. 1750121 ◽  
Author(s):  
Mahdi Reza Khorsandi ◽  
Atefeh Shekofteh
Keyword(s):  
Zero Divisor ◽  
Line Graph ◽  
Line Graphs ◽  
Zero Divisor Graph

In this paper, we study the zero-divisor graph [Formula: see text] of a poset [Formula: see text] and its line graph [Formula: see text]. We characterize all posets whose [Formula: see text] are star, finite complete bipartite or finite. Also, we prove that the diameter of [Formula: see text] is at most 3 while its girth is either 3, 4 or [Formula: see text]. We also characterize [Formula: see text] in terms of their diameter and girth. Finally, we classify all posets [Formula: see text] whose [Formula: see text] are planar.

Line zero divisor graphs

2020 ◽  
pp. 2150154
Author(s):  
Zahra Barati
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Line Graph ◽  
Commutative Rings ◽  
Line Graphs ◽  
Zero Divisor Graph

Let [Formula: see text] be a commutative ring and [Formula: see text] be the zero divisor graph of [Formula: see text]. In this paper, we investigate when the zero divisor graph is a line graph. We completely present all commutative rings which their zero divisor graphs are line graphs. Also, we study when the zero divisor graph is the complement of a line graph.

Some properties about the zero-divisor graphs of quasi-ordered sets

2019 ◽  
Vol 19 (04) ◽  
pp. 2050074
Author(s):  
Junye Ma ◽  
Qingguo Li ◽  
Hui Li
Keyword(s):  
Zero Divisor ◽  
Ordered Set ◽  
Line Graph ◽  
Prime Ideals ◽  
Complete Graphs ◽  
Ordered Sets ◽  
Graph Theoretic ◽  
Zero Divisor Graph ◽  
Complete Bipartite Graphs ◽  
Minimal Prime

In this paper, we study some graph-theoretic properties about the zero-divisor graph [Formula: see text] of a finite quasi-ordered set [Formula: see text] with a least element 0 and its line graph [Formula: see text]. First, we offer a method to find all the minimal prime ideals of a quasi-ordered set. Especially, this method is applicable for a partially ordered set. Then, we completely characterize the diameter and girth of [Formula: see text] by the minimal prime ideals of [Formula: see text]. Besides, we perfectly classify all finite quasi-ordered sets whose zero-divisor graphs are complete graphs, star graphs, complete bipartite graphs, complete [Formula: see text]-partite graphs. We also investigate the planarity of [Formula: see text]. Finally, we obtain the characterization for the line graph [Formula: see text] in terms of its diameter, girth and planarity.

scholarly journals On the Line Graph of the Zero Divisor Graph for the Ring of Gaussian Integers Modulo n

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Khalida Nazzal ◽  
Manal Ghanem
Keyword(s):  
Zero Divisor ◽  
Line Graph ◽  
Domination Number ◽  
The Other ◽  
Graph Invariants ◽  
Gaussian Integers ◽  
Zero Divisor Graph ◽  
Other Hand

Let Γ(ℤn[i]) be the zero divisor graph for the ring of the Gaussian integers modulo n. Several properties of the line graph of Γ(ℤn[i]), L(Γ(ℤn[i])) are studied. It is determined when L(Γ(ℤn[i])) is Eulerian, Hamiltonian, or planer. The girth, the diameter, the radius, and the chromatic and clique numbers of this graph are found. In addition, the domination number of L(Γ(ℤn[i])) is given when n is a power of a prime. On the other hand, several graph invariants for Γ(ℤn[i]) are also determined.

scholarly journals Line Graphs of Monogenic Semigroup Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Nihat Akgunes ◽  
Yasar Nacaroglu ◽  
Sedat Pak
Keyword(s):  
Maximum Degree ◽  
Zero Divisor ◽  
Minimum Degree ◽  
Line Graph ◽  
Domination Number ◽  
Clique Number ◽  
Line Graphs ◽  
Monogenic Semigroup ◽  
Graph Properties ◽  
Graph Parameters

The concept of monogenic semigroup graphs Γ S M is firstly introduced by Das et al. (2013) based on zero divisor graphs. In this study, we mainly discuss the some graph properties over the line graph L Γ S M of Γ S M . In detail, we prove the existence of graph parameters, namely, radius, diameter, girth, maximum degree, minimum degree, chromatic number, clique number, and domination number over L Γ S M .

scholarly journals On the Line Graph for Zero-Divisors of C(X)

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Ghada AlAfifi ◽  
Emad Abu Osba
Keyword(s):  
Hausdorff Space ◽  
Zero Divisor ◽  
Line Graph ◽  
Clique Number ◽  
Completely Regular ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Dominating Number ◽  
Isolated Points

Let be X a completely regular Hausdorff space and let C(X) be the ring of all continuous real valued functions defined on X. In this paper, the line graph for the zero-divisor graph of C(X) is studied. It is shown that this graph is connected with diameter less than or equal to 3 and girth 3. It is shown that this graph is always triangulated and hypertriangulated. It is characterized when the graph is complemented. It is proved that the radius of this graph is 2 if and only if X has isolated points; otherwise, the radius is 3. Bounds for the dominating number and clique number are also found in terms of the density number of X.

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