On the normalized Laplacian spectrum of the comaximal graphs

2021 ◽  
pp. 2250094
Author(s):  
Mojgan Afkhami
Keyword(s):  
Commutative Ring ◽  
Simple Graph ◽  
Laplacian Spectrum ◽  
Induced Subgraph ◽  
Randic Index ◽  
Ring Of Integers ◽  
Laplacian Energy ◽  
Comaximal Graph ◽  
Vertex Set ◽  
Nonzero Identity

Let [Formula: see text] be a commutative ring with nonzero identity. The comaximal graph of [Formula: see text], denoted by [Formula: see text], is a simple graph with 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]. Let [Formula: see text] be an induced subgraph of [Formula: see text] with nonunit elements of [Formula: see text] as vertices. In this paper, we describe the normalized Laplacian spectrum of [Formula: see text], and we determine it for some values of [Formula: see text], where [Formula: see text] is the ring of integers modulo [Formula: see text]. Moreover, we investigate the normalized Laplacian energy and general Randic index of [Formula: see text].

scholarly journals On Laplacian Eigenvalues of the Zero-Divisor Graph Associated to the Ring of Integers Modulo n

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 482
Author(s):  
Bilal A. Rather ◽  
Shariefuddin Pirzada ◽  
Tariq A. Naikoo ◽  
Yilun Shang
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Simple Graph ◽  
Laplacian Spectrum ◽  
Laplacian Eigenvalues ◽  
Ring Of Integers ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set ◽  
Positive Integers

Given a commutative ring R with identity 1≠0, let the set Z(R) denote the set of zero-divisors and let Z*(R)=Z(R)∖{0} be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by Γ(R), is a simple graph whose vertex set is Z*(R) and each pair of vertices in Z*(R) are adjacent when their product is 0. In this article, we find the structure and Laplacian spectrum of the zero-divisor graphs Γ(Zn) for n=pN1qN2, where p<q are primes and N1,N2 are positive integers.

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.

The proof of a conjecture in Jacobson graph of a commutative ring

2015 ◽  
Vol 14 (10) ◽  
pp. 1550107 ◽  
Author(s):  
S. Akbari ◽  
S. Khojasteh ◽  
A. Yousefzadehfard
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Clique Number ◽  
Finite Ring ◽  
Jacobson Graph ◽  
Vertex Set ◽  
Nonzero Identity

Let R be a commutative ring with nonzero identity. The Jacobson graph of R denoted by 𝔍R is a graph with the vertex set R\J(R), and two distinct vertices x, y ∈ V(𝔍R) are adjacent if and only if 1 - xy ∉ U(R), where U(R) is the set of all unit elements of R. Let ω(𝔍R) denote the clique number of 𝔍R. It was conjectured that if [Formula: see text] is a commutative finite ring and (Ri, 𝔪i) is a local ring, for i = 1, …, n, then [Formula: see text], where Fi = Ri/𝔪i, for i = 1, …, n. In this paper, we prove that if R is a commutative ring (not necessarily finite) and R is not a field, then ω(𝔍R) = max 𝔪∈ Max (R) |𝔪| and using this we show that the aforementioned conjecture holds.

Generalized unit and unitary Cayley graphs of finite rings

2019 ◽  
Vol 18 (01) ◽  
pp. 1950006 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
S. Anukumar Kathirvel
Keyword(s):  
Commutative Ring ◽  
Undirected Graph ◽  
Unit Element ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Rings ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Necessary And Sufficient ◽  
Nonzero Identity

Let [Formula: see text] be a finite commutative ring with nonzero identity and [Formula: see text] be the set of all units of [Formula: see text] The graph [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] in which two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if there exists a unit element [Formula: see text] in [Formula: see text] such that [Formula: see text] is a unit in [Formula: see text] In this paper, we obtain degree of all vertices in [Formula: see text] and in turn provide a necessary and sufficient condition for [Formula: see text] to be Eulerian. Also, we give a necessary and sufficient condition for the complement [Formula: see text] to be Eulerian, Hamiltonian and planar.

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.

When is the Jacobson graph projective?

2018 ◽  
Vol 17 (09) ◽  
pp. 1850168
Author(s):  
Atossa Parsapour ◽  
Khadijeh Ahmadjavaheri
Keyword(s):  
Commutative Ring ◽  
Jacobson Radical ◽  
Jacobson Graph ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Nonzero Identity

Let [Formula: see text] be a commutative ring with nonzero identity and [Formula: see text] be the Jacobson radical of [Formula: see text]. The Jacobson graph of [Formula: see text], denoted by [Formula: see text], is a graph with vertex-set [Formula: see text], such that two distinct vertices [Formula: see text] and [Formula: see text] in [Formula: see text] are adjacent if and only if [Formula: see text] is not a unit of [Formula: see text]. The goal in this paper is to list every finite commutative ring [Formula: see text] with nonzero identity (up to isomorphism) such that the graph [Formula: see text] is projective.

On the genus of the graph associated to a commutative ring

2017 ◽  
Vol 09 (05) ◽  
pp. 1750058 ◽  
Author(s):  
K. Selvakumar ◽  
P. Subbulakshmi ◽  
Jafar Amjadi
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Simple Graph ◽  
Vertex Set ◽  
Non Local ◽  
Genus One

Let [Formula: see text] be a commutative ring with identity. We consider a simple graph associated with [Formula: see text], denoted by [Formula: see text], whose vertex set is the set of all non-trivial ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent whenever [Formula: see text] or [Formula: see text]. In this paper, we characterize the commutative Artinian non-local ring [Formula: see text] for which [Formula: see text] has genus one and crosscap one.

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 The radius of unit graphs of rings

2021 ◽  
Vol 6 (10) ◽  
pp. 11508-11515
Author(s):  
Zhiqun Li ◽  
◽  
Huadong Su
Keyword(s):  
Positive Integer ◽  
Simple Graph ◽  
Unit Graph ◽  
Vertex Set ◽  
Ring Extensions ◽  
Nonzero Identity

<abstract><p>Let $ R $ be a ring with nonzero identity. The unit graph of $ R $ is a simple graph whose vertex set is $ R $ itself and two distinct vertices are adjacent if and only if their sum is a unit of $ R $. In this paper, we study the radius of unit graphs of rings. We prove that there exists a ring $ R $ such that the radius of unit graph can be any given positive integer. We also prove that the radius of unit graphs of self-injective rings are 1, 2, 3, $ \infty $. We classify all self-injective rings via the radius of its unit graph. The radius of unit graphs of some ring extensions are also considered.</p></abstract>

scholarly journals More on the annihilator-ideal graph of a commutative ring

2019 ◽  
Vol 18 (08) ◽  
pp. 1950160
Author(s):  
M. J. Nikmehr ◽  
S. M. Hosseini
Keyword(s):  
Commutative Ring ◽  
Sufficient Conditions ◽  
Simple Graph ◽  
Necessary And Sufficient Conditions ◽  
Star Graph ◽  
Annihilator Ideal ◽  
Vertex Set ◽  
Necessary And Sufficient

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 study the affinity between the annihilator-ideal graph and the annihilating-ideal graph [Formula: see text] (a well known graph with the same vertices and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]) associated with [Formula: see text]. All rings whose [Formula: see text] and [Formula: see text] are characterized. Among other results, we obtain necessary and sufficient conditions under which [Formula: see text] is a star graph.

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