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.

On the genus of the k-maximal hypergraph of commutative rings

2019 ◽  
Vol 11 (01) ◽  
pp. 1950010
Author(s):  
K. Selvakumar ◽  
V. C. Amritha
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Local Rings ◽  
Maximal Elements ◽  
Fixed Integer ◽  
Isomorphism Classes ◽  
Vertex Set ◽  
Non Local ◽  
Genus One

Let [Formula: see text] be a commutative ring with identity and [Formula: see text], a fixed integer. Let [Formula: see text] be the set of all [Formula: see text]-maximal elements in [Formula: see text] Associate a [Formula: see text]-maximal hypergraph [Formula: see text] to [Formula: see text] with vertex set [Formula: see text] and for distinct elements [Formula: see text] in [Formula: see text], the set [Formula: see text] is an edge of [Formula: see text] if and only if [Formula: see text] and [Formula: see text] for all [Formula: see text]. In this paper, we determine all isomorphism classes of finite commutative non-local rings with identity whose [Formula: see text]-maximal hypergraph has genus one. Finally, we classify all finite commutative non-local rings [Formula: see text] for which [Formula: see text] is projective.

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).

Finite commutative rings with higher genus unit graphs

2014 ◽  
Vol 14 (01) ◽  
pp. 1550002 ◽  
Author(s):  
Huadong Su ◽  
Kenta Noguchi ◽  
Yiqiang Zhou
Keyword(s):  
Nonnegative Integer ◽  
Commutative Rings ◽  
Orientable Surface ◽  
Simple Graph ◽  
Isomorphism Classes ◽  
Unit Graph ◽  
Higher Genus ◽  
Vertex Set ◽  
Finite Commutative Rings

Let R be a ring with identity. The unit graph of R, denoted by G(R), is a simple graph with vertex set R, and where two distinct vertices x and y are adjacent if and only if x + y is a unit in R. The genus of a simple graph G is the smallest nonnegative integer g such that G can be embedded into an orientable surface Sg. In this paper, we determine all isomorphism classes of finite commutative rings whose unit graphs have genus at most three.

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.

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.

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.

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.

The genus two class of graphs arising from rings

2018 ◽  
Vol 17 (10) ◽  
pp. 1850193 ◽  
Author(s):  
T. Asir
Keyword(s):  
Commutative Ring ◽  
Jacobson Radical ◽  
Commutative Rings ◽  
Isomorphism Classes ◽  
Jacobson Graph ◽  
Vertex Set ◽  
Genus Two

Given a commutative ring [Formula: see text] with identity [Formula: see text], its Jacobson graph [Formula: see text] is defined to be the graph in which the vertex set is [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Here [Formula: see text] denotes the Jacobson radical of [Formula: see text] and [Formula: see text] is the set of unit elements in [Formula: see text]. This paper investigates the genus properties of Jacobson graph. In particular, we determine all isomorphism classes of commutative rings whose Jacobson graph has genus two.

scholarly journals RINGS WHOSE ANNIHILATING-IDEAL GRAPHS HAVE POSITIVE GENUS

2012 ◽  
Vol 11 (03) ◽  
pp. 1250049 ◽  
Author(s):  
F. ALINIAEIFARD ◽  
M. BEHBOODI
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Maximal Ideal ◽  
Artinian Ring ◽  
Commutative Rings ◽  
Gorenstein Ring ◽  
Isomorphism Classes ◽  
Artinian Rings ◽  
Vertex Set ◽  
Positive Genus

Let R be a commutative ring and 𝔸(R) be the set of ideals with nonzero annihilators. The annihilating-ideal graph of R is defined as the graph 𝔸𝔾(R) with the vertex set 𝔸(R)* = 𝔸\{(0)} and two distinct vertices I and J are adjacent if and only if IJ = (0). We investigate commutative rings R whose annihilating-ideal graphs have positive genus γ(𝔸𝔾(R)). It is shown that if R is an Artinian ring such that γ(𝔸𝔾(R)) < ∞, then either R has only finitely many ideals or (R, 𝔪) is a Gorenstein ring with maximal ideal 𝔪 and v.dimR/𝔪𝔪/𝔪2= 2. Also, for any two integers g ≥ 0 and q > 0, there are only finitely many isomorphism classes of Artinian rings R satisfying the conditions: (i) γ(𝔸𝔾(R)) = g and (ii) |R/𝔪| ≤ q for every maximal ideal 𝔪 of R. Also, it is shown that if R is a non-domain Noetherian local ring such that γ(𝔸𝔾(R)) < ∞, then either R is a Gorenstein ring or R is an Artinian ring with only finitely many ideals.

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)).

Sign in / Sign up

Export Citation Format

Share Document