Crosscap two of class of graphs from commutative rings

2021 ◽  
Author(s):  
S. Karthik ◽  
S. N. Meera ◽  
K. Selvakumar
Keyword(s):  
Commutative Ring ◽  
Undirected Graph ◽  
Commutative Rings ◽  
Zero Divisors ◽  
Essential Ideal ◽  
Vertex Set ◽  
Annihilator Graph ◽  
Finite Commutative Rings

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of all nonzero zero-divisors of [Formula: see text]. The annihilator graph of commutative ring [Formula: see text] is the simple undirected graph [Formula: see text] with vertices [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. The essential graph of [Formula: see text] is defined as the graph [Formula: see text] 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] is an essential ideal. In this paper, we classify all finite commutative rings with identity whose annihilator graph and essential graph have crosscap two.

Finite commutative ring with genus two essential graph

2018 ◽  
Vol 17 (07) ◽  
pp. 1850121
Author(s):  
K. Selvakumar ◽  
M. Subajini ◽  
M. J. Nikmehr
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Zero Divisors ◽  
Essential Ideal ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Genus Two ◽  
Finite Commutative Rings

Let [Formula: see text] be a commutative ring with identity and let [Formula: see text] be the set of zero-divisors of [Formula: see text]. The essential graph of [Formula: see text] is defined as the graph [Formula: see text] 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] is an essential ideal. In this paper, we classify all finite commutative rings with identity for which the genus of [Formula: see text] is two.

The metric dimension of annihilator graphs of commutative rings

2019 ◽  
Vol 19 (05) ◽  
pp. 2050089
Author(s):  
V. Soleymanivarniab ◽  
A. Tehranian ◽  
R. Nikandish
Keyword(s):  
Commutative Ring ◽  
Undirected Graph ◽  
Commutative Rings ◽  
Metric Dimension ◽  
Zero Divisors ◽  
Vertex Set ◽  
Annihilator Graph ◽  
Nonzero Identity

Let [Formula: see text] be a commutative ring with nonzero identity. The annihilator graph of [Formula: see text], denoted by [Formula: see text], is the (undirected) graph whose vertex set is the set of all nonzero zero-divisors of [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 study the metric dimension of annihilator graphs associated with commutative rings and some metric dimension formulae for annihilator graphs are given.

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 When the annihilator graph of a commutative ring is planar or toroidal?

2020 ◽  
Vol 24 (2) ◽  
pp. 281-290
Author(s):  
Moharram Bakhtyiari ◽  
Reza Nikandish ◽  
Mohammad Javad Nikmehr
Keyword(s):  
Commutative Ring ◽  
Undirected Graph ◽  
Zero Divisors ◽  
Vertex Set ◽  
Annihilator Graph

Let R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. The annihilator graph of R is defined as the undirected graph AG(R) with the vertex set Z(R)* = Z(R) \ {0}, and two distinct vertices x and y are adjacent if and only if  ann_R(xy) \neq ann_R(x) \cup ann_R(y). In this paper, all rings whose annihilator graphs can be embedded on the plane or torus are classified.

On perfect annihilator graphs of commutative rings

2020 ◽  
Vol 12 (05) ◽  
pp. 2050060
Author(s):  
Sh. Ebrahimi ◽  
A. Tehranian ◽  
R. Nikandish
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Zero Divisors ◽  
Vertex Set ◽  
Annihilator Graph ◽  
Vast Range

Let [Formula: see text] be a commutative ring with identity, and let [Formula: see text] be the set of zero-divisors of [Formula: see text]. The annihilator graph of [Formula: see text] is defined as the graph [Formula: see text] 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 study the perfectness of annihilator graphs of a vast range of rings. Indeed, it is shown that if [Formula: see text] is reduced with finitely many minimal primes or nonreduced, then [Formula: see text] is perfect.

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).

PLANAR, OUTERPLANAR, AND RING GRAPH OF THE COZERO-DIVISOR GRAPH OF A FINITE COMMUTATIVE RING

2012 ◽  
Vol 11 (06) ◽  
pp. 1250103 ◽  
Author(s):  
MOJGAN AFKHAMI ◽  
KAZEM KHASHYARMANESH
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Ring Graph ◽  
Nonzero Identity ◽  
Finite Commutative Rings

Let R be a commutative ring with nonzero identity. The cozero-divisor graph of R, denoted by Γ′(R), is a graph with vertex-set W*(R), which is the set of all nonzero and non-unit elements of R, and two distinct vertices a and b in W*(R) are adjacent if and only if a ∉ Rb and b ∉ Ra. In this paper, we characterize all finite commutative rings R such that Γ′(R) is planar, outerplanar or ring graph.

Nilpotent graphs of genus one

2014 ◽  
Vol 06 (03) ◽  
pp. 1450037
Author(s):  
R. Kala ◽  
S. Kavitha
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Isomorphism Classes ◽  
Vertex Set ◽  
Genus One ◽  
Finite Commutative Rings

Let R be a commutative ring with identity. The nilpotent graph of R, denoted by ΓN(R), is a graph with vertex set [Formula: see text], and two vertices x and y are adjacent if and only if xy is nilpotent, where [Formula: see text] is nilpotent, for some y ∈ R*}. In this paper, we determine all isomorphism classes of finite commutative rings with identity whose ΓN(R) has genus one.

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 .

scholarly journals Maximal Ideal Graph of Commutative Rings

2020 ◽  
pp. 2070-2076
Author(s):  
F. H. Abdulqadr
Keyword(s):  
Commutative Ring ◽  
Maximal Ideal ◽  
Undirected Graph ◽  
Commutative Rings ◽  
Vertex Set

In this paper, we introduce and study the notion of the maximal ideal graph of a commutative ring with identity. Let R be a commutative ring with identity. The maximal ideal graph of R, denoted by MG(R), is the undirected graph with vertex set, the set of non-trivial ideals of R, where two vertices I1 and I2 are adjacent if I1 I2 and I1+I2 is a maximal ideal of R. We explore some of the properties and characterizations of the graph.

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