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.

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.

Some Results on the Domination Number of a Zero-divisor Graph

2014 ◽  
Vol 57 (3) ◽  
pp. 573-578 ◽  
Author(s):  
Sima Kiani ◽  
Hamid Reza Maimani ◽  
Reza Nikandish
Keyword(s):  
Noetherian Ring ◽  
Zero Divisor ◽  
Domination Number ◽  
Total Domination ◽  
Ore Extension ◽  
Commutative Noetherian Ring ◽  
Zero Divisor Graph ◽  
Domination Numbers

AbstractIn this paper, we investigate the domination, total domination, and semi-total domination numbers of a zero-divisor graph of a commutative Noetherian ring. Also, some relations between the domination numbers of Γ(R/I) and Γ1(R), and the domination numbers of Γ(R) and Γ(R[x, α, δ]), where R[x, α, δ] is the Ore extension of R, are studied.

Zero-divisor graphs of matrices over lattices

2016 ◽  
Vol 08 (04) ◽  
pp. 1650060 ◽  
Author(s):  
Anagha Khiste ◽  
Vinayak Joshi
Keyword(s):  
Chromatic Number ◽  
Zero Divisor ◽  
Domination Number ◽  
Basic Properties ◽  
Zero Divisor Graph ◽  
Element Chain ◽  
Text Element ◽  
Triangulated Graph

In this paper, we study basic properties such as connectivity, diameter and girth of the zero-divisor graph [Formula: see text] of [Formula: see text] matrices over a lattice [Formula: see text] with 0. Further, we consider the zero-divisor graph [Formula: see text] of [Formula: see text] matrices over an [Formula: see text]-element chain [Formula: see text]. We determine the number of vertices, degree of each vertex, domination number and edge chromatic number of [Formula: see text]. Also, we show that Beck’s Conjecture is true for [Formula: see text]. Further, we prove that [Formula: see text] is hyper-triangulated graph.

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 Matching Number, Independence Number, and Covering Vertex Number of Γ(Zn)

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 49
Author(s):  
Eman AbuHijleh ◽  
Mohammad Abudayah ◽  
Omar Alomari ◽  
Hasan Al-Ezeh
Keyword(s):  
Zero Divisor ◽  
Independence Number ◽  
Independent Set ◽  
Covering Problem ◽  
Matching Number ◽  
Graph Invariants ◽  
Vertex Number ◽  
Zero Divisor Graph ◽  
Vertex Covering ◽  
Number Problem

Graph invariants are the properties of graphs that do not change under graph isomorphisms, the independent set decision problem, vertex covering problem, and matching number problem are known to be NP-Hard, and hence it is not believed that there are efficient algorithms for solving them. In this paper, the graph invariants matching number, vertex covering number, and independence number for the zero-divisor graph over the rings Z p k and Z p k q r are determined in terms of the sets S p i and S p i q j respectively. Accordingly, a formula in terms of p , q , k , and r, with n = p k , n = p k q r is provided.

Zero Divisor Graph for the Ring of Gaussian Integers Modulon

2008 ◽  
Vol 36 (10) ◽  
pp. 3865-3877 ◽  
Author(s):  
Emad Abu Osba ◽  
Salah Al-Addasi ◽  
Nafiz Abu Jaradeh
Keyword(s):  
Zero Divisor ◽  
Gaussian Integers ◽  
Zero Divisor Graph

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.

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

2011 ◽  
Vol 53 (2) ◽  
pp. 391-399 ◽  
Author(s):  
EMAD ABU OSBA ◽  
SALAH AL-ADDASI ◽  
BASEM AL-KHAMAISEH
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Zero Divisor ◽  
Gaussian Integers ◽  
Zero Divisor Graph

AbstractThis paper is a continuation for the study of the zero-divisor graph for the ring of Gaussian integers modulo n, Γ(ℤn[i]) in [8] (Emad Abu Osba, Salah Al-Addasi and Nafez Abu Jaradeh. Zero divisor graph for the ring of Gaussin integers modulo n. Comm. Algebra 36(10) (2008), 3865–3877). It is investigated, when is Γ(ℤn[i]) locally H, Hamiltonian or bipartite graph? A full characterisation for the chromatic number and the radius is also given.

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