scholarly journals Domination number in the annihilating-ideal graphs of commutative rings

2015 ◽  
Vol 97 (111) ◽  
pp. 225-231 ◽  
Author(s):  
Reza Nikandish ◽  
Hamid Maimani ◽  
Sima Kiani
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Domination Number ◽  
Commutative Rings ◽  
Vertex Set ◽  
Domination Numbers

Let R be a commutative ring with identity and A(R) be the set of ideals with nonzero annihilator. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R)* = A(R)\{0} and two distinct vertices I and J are adjacent if and only if IJ = 0. In this paper, we study the domination number of AG(R) and some connections between the domination numbers of annihilating-ideal graphs and zero-divisor graphs are given.

Annihilating-ideal graphs of commutative rings

2019 ◽  
Vol 13 (07) ◽  
pp. 2050121
Author(s):  
M. Aijaz ◽  
S. Pirzada
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Metric Dimension ◽  
Maximal Ideals ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with unity [Formula: see text]. The annihilating-ideal graph of [Formula: see text], denoted by [Formula: see text], is defined to be the graph with vertex set [Formula: see text] — the set of non-zero annihilating ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] adjacent if and only if [Formula: see text]. Some connections between annihilating-ideal graphs and zero divisor graphs are given. We characterize the prime ideals (or equivalently maximal ideals) of [Formula: see text] in terms of their degrees as vertices of [Formula: see text]. We also obtain the metric dimension of annihilating-ideal graphs of commutative rings.

scholarly journals Metric and upper dimension of zero divisor graphs associated to commutative rings

2020 ◽  
Vol 12 (1) ◽  
pp. 84-101 ◽  
Author(s):  
S. Pirzada ◽  
M. Aijaz
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Metric Dimension ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set ◽  
Finite Commutative Rings

AbstractLet R be a commutative ring with Z*(R) as the set of non-zero zero divisors. The zero divisor graph of R, denoted by Γ(R), is the graph whose vertex set is Z*(R), where two distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we investigate the metric dimension dim(Γ(R)) and upper dimension dim+(Γ(R)) of zero divisor graphs of commutative rings. For zero divisor graphs Γ(R) associated to finite commutative rings R with unity 1 ≠ 0, we conjecture that dim+(Γ(R)) = dim(Γ(R)), with one exception that {\rm{R}} \cong \Pi {\rm\mathbb{Z}}_2^{\rm{n}}, n ≥ 4. We prove that this conjecture is true for several classes of rings. We also provide combinatorial formulae for computing the metric and upper dimension of zero divisor graphs of certain classes of commutative rings besides giving bounds for the upper dimension of zero divisor graphs of rings.

scholarly journals Planar, Outerplanar, and Toroidal Graphs of the Generalized Zero-Divisor Graph of Commutative Rings

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Abdulaziz M. Alanazi ◽  
Mohd Nazim ◽  
Nadeem Ur Rehman
Keyword(s):  
Zero Divisor ◽  
Domination Number ◽  
Commutative Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Nonzero Intersection ◽  
Toroidal Graphs ◽  
Vertex Set ◽  
Generalized Zero ◽  
Finite Commutative Rings

Let A be a commutative ring with unity and let set of all zero divisors of A be denoted by Z A . An ideal ℐ of the ring A is said to be essential if it has a nonzero intersection with every nonzero ideal of A . It is denoted by ℐ ≤ e A . The generalized zero-divisor graph denoted by Γ g A is an undirected graph with vertex set Z A ∗ (set of all nonzero zero-divisors of A ) and two distinct vertices x 1 and x 2 are adjacent if and only if ann x 1 + ann x 2 ≤ e A . In this paper, first we characterized all the finite commutative rings A for which Γ g A is isomorphic to some well-known graphs. Then, we classify all the finite commutative rings A for which Γ g A is planar, outerplanar, or toroidal. Finally, we discuss about the domination number of Γ g A .

Ideal-Based k-Zero-Divisor Hypergraph of Commutative Rings

2021 ◽  
Vol 28 (04) ◽  
pp. 655-672
Author(s):  
K. Selvakumar ◽  
M. Subajini
Keyword(s):  
Commutative Ring ◽  
Finite Fields ◽  
Zero Divisor ◽  
Necessary Conditions ◽  
Commutative Rings ◽  
Nonzero Ideal ◽  
Fixed Integer ◽  
Zero Divisors ◽  
Vertex Set ◽  
The Ideal

Let [Formula: see text] be a commutative ring, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] a fixed integer. The ideal-based [Formula: see text]-zero-divisor hypergraph [Formula: see text] of [Formula: see text] has vertex set [Formula: see text], the set of all ideal-based [Formula: see text]-zero-divisors of [Formula: see text], and for distinct elements [Formula: see text] in [Formula: see text], the set [Formula: see text] is an edge in [Formula: see text] if and only if [Formula: see text] and the product of the elements of any [Formula: see text]-subset of [Formula: see text] is not in [Formula: see text]. In this paper, we show that [Formula: see text] is connected with diameter at most 4 provided that [Formula: see text] for all ideal-based 3-zero-divisor hypergraphs. Moreover, we find the chromatic number of [Formula: see text] when [Formula: see text] is a product of finite fields. Finally, we find some necessary conditions for a finite ring [Formula: see text] and a nonzero ideal [Formula: see text] of [Formula: see text] to have [Formula: see text] planar.

On locating numbers and codes of zero divisor graphs associated with commutative rings

2015 ◽  
Vol 15 (01) ◽  
pp. 1650014 ◽  
Author(s):  
Rameez Raja ◽  
S. Pirzada ◽  
Shane Redmond
Keyword(s):  
Commutative Ring ◽  
Connected Graph ◽  
Zero Divisor ◽  
Domination Number ◽  
Clique Number ◽  
Commutative Rings ◽  
Equivalence Relations ◽  
Finite Product ◽  
Positive Integers ◽  
The Relationship

Let R be a commutative ring with identity and let G(V, E) be a graph. The locating number of the graph G(V, E) denoted by loc (G) is the cardinality of the minimal locating set W ⊆ V(G). To get the loc (G), we assign locating codes to the vertices V(G)∖W of G in such a way that every two vertices get different codes. In this paper, we consider the ratio of loc (G) to |V(G)| and show that there is a finite connected graph G with loc (G)/|V(G)| = m/n, where m < n are positive integers. We examine two equivalence relations on the vertices of Γ(R) and the relationship between locating sets and the cut vertices of Γ(R). Further, we obtain bounds for the locating number in zero-divisor graphs of a commutative ring and discuss the relation between locating number, domination number, clique number and chromatic number of Γ(R). We also investigate the locating number in Γ(R) when R is a finite product of rings.

DOMINATION IN THE TOTAL GRAPH ON ℤn

2011 ◽  
Vol 03 (04) ◽  
pp. 413-421 ◽  
Author(s):  
T. TAMIZH CHELVAM ◽  
T. ASIR
Keyword(s):  
Commutative Ring ◽  
Prime Divisor ◽  
Undirected Graph ◽  
Domination Number ◽  
Total Graph ◽  
Zero Divisors ◽  
Vertex Set ◽  
Domination Numbers

For a commutative ring R, let Z(R) be its set of zero-divisors. The total graph of R, denoted by TΓ(R), is the undirected graph with vertex set R, and for distinct x, y ∈ R, the vertices x and y are adjacent if and only if x + y ∈ Z(R). Tamizh Chelvam and Asir studied about the domination in the total graph of a commutative ring R. In particular, it was proved that the domination number γ(TΓ(ℤn)) = p1 where p1 is the smallest prime divisor of n. In this paper, we characterize all the γ-sets in TΓ(ℤn). Also, we obtain the values of other domination parameters like independent, total and perfect domination numbers of the total graph on ℤn.

scholarly journals Total and Cototal Domination Number of Some Zero Divisor Graph

2019 ◽  
Vol 8 (3S2) ◽  
pp. 950-952
Keyword(s):  
Graph Theory ◽  
Commutative Ring ◽  
Direct Product ◽  
Zero Divisor ◽  
Domination Number ◽  
Commutative Rings ◽  
Ring Theory ◽  
Research Topics ◽  
Zero Divisor Graph ◽  
Domination In Graphs

The concept of zero divisor graphs of a commutative ring leads to various research topics since it relates both Ring theory and Graph theory. Domination in graphs has its own development in each field it enters. In this paper, We have given few results on Total and cototal domination number of some zero divisor graph on direct product of commutative rings.

THE INTERSECTION GRAPH OF GAMMA SETS IN THE TOTAL GRAPH OF A COMMUTATIVE RING-II

2013 ◽  
Vol 12 (04) ◽  
pp. 1250199 ◽  
Author(s):  
T. ASIR ◽  
T. TAMIZH CHELVAM
Keyword(s):  
Commutative Ring ◽  
Independence Number ◽  
Clique Number ◽  
Commutative Rings ◽  
Intersection Graph ◽  
Vertex Connectivity ◽  
Total Graph ◽  
Graph Theoretic ◽  
Vertex Set ◽  
Finite Commutative Rings

The intersection graph ITΓ(R) of gamma sets in the total graph TΓ(R) of a commutative ring R, is the undirected graph with vertex set as the collection of all γ-sets in the total graph of R and two distinct vertices u and v are adjacent if and only if u ∩ v ≠ ∅. Tamizh Chelvam and Asir [The intersection graph of gamma sets in the total graph I, to appear in J. Algebra Appl.] studied about ITΓ(R) where R is a commutative Artin ring. In this paper, we continue our interest on ITΓ(R) and actually we study about Eulerian, Hamiltonian and pancyclic nature of ITΓ(R). Further, we focus on certain graph theoretic parameters of ITΓ(R) like the independence number, the clique number and the connectivity of ITΓ(R). Also, we obtain both vertex and edge chromatic numbers of ITΓ(R). In fact, it is proved that if R is a finite commutative ring, then χ(ITΓ(R)) = ω(ITΓ(R)). Having proved that ITΓ(R) is weakly perfect for all finite commutative rings, we further characterize all finite commutative rings for which ITΓ(R) is perfect. In this sequel, we characterize all commutative Artin rings for which ITΓ(R) is of class one (i.e. χ′(ITΓ(R)) = Δ(ITΓ(R))). Finally, it is proved that the vertex connectivity and edge connectivity of ITΓ(R) are equal to the degree of any vertex in ITΓ(R).

ON RINGS R WHOSE GRAPHS Γ(R) SATISFY CONDITION (Kp)

2011 ◽  
Vol 10 (04) ◽  
pp. 665-674
Author(s):  
LI CHEN ◽  
TONGSUO WU
Keyword(s):  
Prime Number ◽  
Commutative Ring ◽  
Complete Graph ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Satisfy Condition ◽  
Induced Subgraph ◽  
Zero Divisor Graph ◽  
Ring Graph ◽  
Finite Commutative Rings

Let p be a prime number. Let G = Γ(R) be a ring graph, i.e. the zero-divisor graph of a commutative ring R. For an induced subgraph H of G, let CG(H) = {z ∈ V(G) ∣N(z) = V(H)}. Assume that in the graph G there exists an induced subgraph H which is isomorphic to the complete graph Kp-1, a vertex c ∈ CG(H), and a vertex z such that d(c, z) = 3. In this paper, we characterize the finite commutative rings R whose graphs G = Γ(R) have this property (called condition (Kp)).

The Weakly Nilpotent Graph of a Commutative Ring

2017 ◽  
Vol 60 (2) ◽  
pp. 319-328
Author(s):  
Soheila Khojasteh ◽  
Mohammad Javad Nikmehr
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Disjoint Union ◽  
Independence Number ◽  
Artinian Ring ◽  
Clique Number ◽  
Commutative Rings ◽  
Unicyclic Graph ◽  
Nilpotent Elements ◽  
Vertex Set

AbstractLet R be a commutative ring with non-zero identity. In this paper, we introduce theweakly nilpotent graph of a commutative ring. The weakly nilpotent graph of R denoted by Γw(R) is a graph with the vertex set R* and two vertices x and y are adjacent if and only if x y ∊ N(R)*, where R* = R \ {0} and N(R)* is the set of all non-zero nilpotent elements of R. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if Γw(R) is a forest, then Γw(R) is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of Γw(R). Among other results, we show that for an Artinian ring R, Γw(R) is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam . Finally, we characterize all commutative rings R for which is a cycle, where is the complement of the weakly nilpotent graph of R.

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