scholarly journals Complement of the generalized total graph of Zn

Filomat ◽  
2019 ◽  
Vol 33 (18) ◽  
pp. 6103-6113 ◽  
Author(s):  
Chelvam Tamizh ◽  
M. Balamurugan
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Independence Number ◽  
Domination Number ◽  
Simple Graph ◽  
Total Graph ◽  
Vertex Set

Let R be a commutative ring with identity and H be a nonempty proper multiplicative prime subset of R. The generalized total graph of R is the (undirected) simple graph ?GTH(R) with all elements of R as the vertex set and two distinct vertices x and y are adjacent if and only if x + y ? H. The complement of the generalized total graph ?GTH(R) of R is the (undirected) simple graphwith vertex set R and two distinct vertices x and y are adjacent if and only if x + y < H. In this paper, we investigate certain domination properties of ?GTH(R). In particular, we obtain the domination number, independence number and a characterization for -sets in ?GTP(Zn) where P is a prime ideal of Zn. Further, we discuss properties like Eulerian, Hamiltonian, planarity, and toroidality of GTP(Zn).

SOME CRITERIA FOR THE FINITENESS OF COZERO-DIVISOR GRAPHS

2013 ◽  
Vol 12 (08) ◽  
pp. 1350056 ◽  
Author(s):  
S. AKBARI ◽  
S. KHOJASTEH
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Noetherian Ring ◽  
Maximum Degree ◽  
Independence Number ◽  
Domination Number ◽  
Finite Ring ◽  
Vertex Set ◽  
Isolated Vertex ◽  
The Ideal

Let R be a commutative ring with unity. The cozero-divisor graph of R denoted by Γ'(R) is a graph with the vertex set W*(R), where W*(R) is the set of all nonzero and non-unit elements of R, and two distinct vertices a and b are adjacent if and only if a ∉ Rb and b ∉ Ra, where Rc is the ideal generated by the element c in R. Let α(Γ'(R)) and γ(Γ'(R)) denote the independence number and the domination number of Γ'(R), respectively. In this paper, we prove that if α(Γ'(R)) is finite, then R is Artinian if and only if R is Noetherian. Also, we prove that if α(Γ'(R)) is finite, then R/P is finite, for every prime ideal P. Moreover, we prove that if R is a Noetherian ring, γ(Γ'(R)) is finite and Γ'(R) has at least one isolated vertex, then J(R) = N(R). We show that if R is a commutative Noetherian local ring, γ(Γ'(R)) is finite and Γ'(R) has at least one isolated vertex, then R is a finite ring. Among other results, we prove that if R is a commutative ring and the maximum degree of Γ'(R) is finite and positive, then R is a finite ring.

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).

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

2013 ◽  
Vol 12 (04) ◽  
pp. 1250198 ◽  
Author(s):  
T. TAMIZH CHELVAM ◽  
T. ASIR
Keyword(s):  
Commutative Ring ◽  
Domination Number ◽  
Intersection Graph ◽  
Total Graph ◽  
Graph Theoretic ◽  
Zero Divisors ◽  
Artin Ring ◽  
Transitive Property ◽  
Vertex Set ◽  
Vertex Transitive

Let R be a commutative ring and Z(R) be its set of all zero-divisors. Anderson and Badawi [The total graph of a commutative ring, J. Algebra320 (2008) 2706–2719] introduced the total graph of R, denoted by TΓ(R), as the undirected graph with vertex set R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). Tamizh Chelvam and Asir [Domination in the total graph of a commutative ring, to appear in J. Combin. Math. Combin. Comput.] obtained the domination number of the total graph and studied certain other domination parameters of TΓ(R) where R is a commutative Artin ring. The intersection graph of gamma sets in TΓ(R) is denoted by ITΓ(R). Tamizh Chelvam and Asir [Intersection graph of gamma sets in the total graph, Discuss. Math. Graph Theory32 (2012) 339–354, doi:10.7151/dmgt.1611] initiated a study about the intersection graph ITΓ (ℤn) of gamma sets in TΓ(ℤn). In this paper, we study about ITΓ(R), where R is a commutative Artin ring. Actually we investigate the interplay between graph-theoretic properties of ITΓ(R) and ring-theoretic properties of R. At the first instance, we prove that diam (ITΓ(R)) ≤ 2 and gr (ITΓ(R)) ≤ 4. Also some characterization results regarding completeness, bipartite, cycle and chordal nature of ITΓ(R) are given. Further, we discuss about the vertex-transitive property of ITΓ(R). At last, we obtain all commutative Artin rings R for which ITΓ(R) is either planar or toroidal or genus two.

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.

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.

THE JACOBSON GRAPH OF COMMUTATIVE RINGS

2012 ◽  
Vol 12 (03) ◽  
pp. 1250179 ◽  
Author(s):  
A. AZIMI ◽  
A. ERFANIAN ◽  
M. FARROKHI D. G.
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Independence Number ◽  
Commutative Rings ◽  
Dominating Number ◽  
Jacobson Graph ◽  
Vertex Set ◽  
Numerical Invariants ◽  
Nonzero Identity

Let R be a commutative ring with nonzero identity. Then the Jacobson graph of R, denoted by 𝔍R, is defined as a graph with vertex set R\J(R) such that two distinct vertices x and y are adjacent if and only if 1 - xy is not a unit of R. We obtain some graph theoretical properties of 𝔍R including its connectivity, planarity and perfectness and we compute some of its numerical invariants, namely diameter, girth, dominating number, independence number and vertex chromatic number and give an estimate for its edge chromatic number.

scholarly journals On k-rainbow domination in middle graphs

2021 ◽  
Author(s):  
Kijung Kim
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Simple Graph ◽  
Upper And Lower Bounds ◽  
Matching Number ◽  
Domatic Number ◽  
Rainbow Domination ◽  
Vertex Set ◽  
Middle Graph ◽  
Dominating Function

Let $G$ be a finite simple graph with vertex set $V(G)$ and edge set $E(G)$. A function $f : V(G) \rightarrow \mathcal{P}(\{1, 2, \dotsc, k\})$ is a \textit{$k$-rainbow dominating function} on $G$ if for each vertex $v \in V(G)$ for which $f(v)= \emptyset$, it holds that $\bigcup_{u \in N(v)}f(u) = \{1, 2, \dotsc, k\}$. The weight of a $k$-rainbow dominating function is the value $\sum_{v \in V(G)}|f(v)|$. The \textit{$k$-rainbow domination number} $\gamma_{rk}(G)$ is the minimum weight of a $k$-rainbow dominating function on $G$. In this paper, we initiate the study of $k$-rainbow domination numbers in middle graphs. We define the concept of a middle $k$-rainbow dominating function, obtain some bounds related to it and determine the middle $3$-rainbow domination number of some classes of graphs. We also provide upper and lower bounds for the middle $3$-rainbow domination number of trees in terms of the matching number. In addition, we determine the $3$-rainbow domatic number for the middle graph of paths and cycles.

A generalized ideal-based zero-divisor graph

2015 ◽  
Vol 14 (06) ◽  
pp. 1550079 ◽  
Author(s):  
M. J. Nikmehr ◽  
S. Khojasteh
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Chromatic Number ◽  
Zero Divisor ◽  
Undirected Graph ◽  
Clique Number ◽  
Proper Ideal ◽  
Graph Structure ◽  
Zero Divisor Graph ◽  
Vertex Set

Let R be a commutative ring with identity, I its proper ideal and M be a unitary R-module. In this paper, we introduce and study a kind of graph structure of an R-module M with respect to proper ideal I, denoted by ΓI(RM) or simply ΓI(M). It is the (undirected) graph with the vertex set M\{0} and two distinct vertices x and y are adjacent if and only if [x : M][y : M] ⊆ I. Clearly, the zero-divisor graph of R is a subgraph of Γ0(R); this is an important result on the definition. We prove that if ann R(M) ⊆ I and H is the subgraph of ΓI(M) induced by the set of all non-isolated vertices, then diam (H) ≤ 3 and gr (ΓI(M)) ∈ {3, 4, ∞}. Also, we prove that if Spec (R) and ω(Γ Nil (R)(M)) are finite, then χ(Γ Nil (R)(M)) ≤ ∣ Spec (R)∣ + ω(Γ Nil (R)(M)). Moreover, for a secondary R-module M and prime ideal P, we determine the chromatic number and the clique number of ΓP(M), where ann R(M) ⊆ P. Among other results, it is proved that for a semisimple R-module M with ann R(M) ⊆ I, ΓI(M) is a forest if and only if ΓI(M) is a union of isolated vertices or a star.

When the annihilator-ideal graph is planar or complete bipartite graph

2019 ◽  
Vol 12 (02) ◽  
pp. 1950024
Author(s):  
M. J. Nikmehr ◽  
S. M. Hosseini
Keyword(s):  
Commutative Ring ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Simple Graph ◽  
Annihilator Ideal ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of ideals of [Formula: see text] with nonzero annihilator. The annihilator-ideal graph of [Formula: see text], denoted by [Formula: see text], is a simple graph with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we present some results on the bipartite, complete bipartite, outer planar and unicyclic of the annihilator-ideal graphs of a commutative ring. Among other results, bipartite annihilator-ideal graphs of rings are characterized. Also, we investigate planarity of the annihilator-ideal graph and classify rings whose annihilator-ideal graph is planar.

Outer-paired domination in graphs

2020 ◽  
Vol 12 (06) ◽  
pp. 2050072
Author(s):  
A. Mahmoodi ◽  
L. Asgharsharghi
Keyword(s):  
Perfect Matching ◽  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Minimum Cardinality ◽  
Sharp Bounds ◽  
Paired Domination Number ◽  
Vertex Set ◽  
Paired Domination ◽  
Domination In Graphs

Let [Formula: see text] be a simple graph with vertex set [Formula: see text] and edge set [Formula: see text]. An outer-paired dominating set [Formula: see text] of a graph [Formula: see text] is a dominating set such that the subgraph induced by [Formula: see text] has a perfect matching. The outer-paired domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of an outer-paired dominating set of [Formula: see text]. In this paper, we study the outer-paired domination number of graphs and present some sharp bounds concerning the invariant. Also, we characterize all the trees with [Formula: see text].

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