Spectra of Boolean Graphs Over Finite Fields of Characteristic Two

2019 ◽  
Vol 63 (1) ◽  
pp. 58-65
Author(s):  
D. Scott Dillery ◽  
John D. LaGrange
Keyword(s):  
Number Theory ◽  
Finite Fields ◽  
Adjacency Matrix ◽  
Zero Divisor ◽  
Algebraic Closure ◽  
Simple Graph ◽  
Boolean Ring ◽  
Zero Divisor Graph ◽  
Finite Commutative Ring ◽  
Characteristic Two

AbstractWith entries of the adjacency matrix of a simple graph being regarded as elements of $\mathbb{F}_{2}$, it is proved that a finite commutative ring $R$ with $1\neq 0$ is a Boolean ring if and only if either $R\in \{\mathbb{F}_{2},\mathbb{F}_{2}\times \mathbb{F}_{2}\}$ or the eigenvalues (in the algebraic closure of $\mathbb{F}_{2}$) corresponding to the zero-divisor graph of $R$ are precisely the elements of $\mathbb{F}_{4}\setminus \{0\}$ . This is achieved by observing a way in which algebraic behavior in a Boolean ring is encoded within Pascal’s triangle so that computations can be carried out by appealing to classical results from number theory.

On zero-divisor graphs of commutative rings without identity

2019 ◽  
Vol 19 (12) ◽  
pp. 2050226 ◽  
Author(s):  
G. Kalaimurugan ◽  
P. Vignesh ◽  
T. Tamizh Chelvam
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Zero Divisor Graph ◽  
Finite Commutative Ring ◽  
Free Order ◽  
Finite Commutative Rings

Let [Formula: see text] be a finite commutative ring without identity. In this paper, we characterize all finite commutative rings without identity, whose zero-divisor graphs are unicyclic, claw-free and tree. Also, we obtain all finite commutative rings without identity and of cube-free order for which the corresponding zero-divisor graph is toroidal.

Divisor graph of the complement of Γ(R)

2019 ◽  
Vol 12 (04) ◽  
pp. 1950057
Author(s):  
Ravindra Kumar ◽  
Om Prakash
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Integral Domain ◽  
Zero Divisor ◽  
Local Rings ◽  
Zero Divisor Graph ◽  
Finite Commutative Ring

Let [Formula: see text] be the complement of the zero-divisor graph of a finite commutative ring [Formula: see text]. In this paper, we provide the answer of the question (ii) raised by Osba and Alkam in [11] and prove that [Formula: see text] is a divisor graph if [Formula: see text] is a local ring. It is shown that when [Formula: see text] is a product of two local rings, then [Formula: see text] is a divisor graph if one of them is an integral domain. Further, if [Formula: see text], then [Formula: see text] is a divisor graph.

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.

scholarly journals A generalization of the zero-divisor graph for modules

2019 ◽  
Vol 106 (120) ◽  
pp. 39-46
Author(s):  
Katayoun Nozari ◽  
Shiroyeh Payrovi
Keyword(s):  
Commutative Ring ◽  
Complete Graph ◽  
Zero Divisor ◽  
Simple Graph ◽  
Zero Divisor Graph ◽  
Maximal Ideals

Let R be a commutative ring and M a Noetherian R-module. The zero-divisor graph of M, denoted by ?(M), is an undirected simple graph whose vertices are the elements of ZR(M)\AnnR(M) and two distinct vertices a and b are adjacent if and only if abM = 0. In this paper, we study diameter and girth of ?(M). We show that the zero-divisor graph of M has a universal vertex in ZR(M)\r(AnnR(M)) if and only if R = ?Z2?R? and M = Z2?M?, where M? is an R?-module. Moreover, we show that if ?(M) is a complete graph, then one of the following statements is true: (i) AssR(M) = {m1,m2}, where m1,m2 are maximal ideals of R. (ii) AssR(M) = {p}, where p 2 ? AnnR(M). (iii) AssR(M) = {p}, where p 3 ? AnnR(M).

scholarly journals On the Genus of the Zero-Divisor Graph of Zn

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Huadong Su ◽  
Pailing Li
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Orientable Surface ◽  
Simple Graph ◽  
Ring Of Integers ◽  
Zero Divisor Graph ◽  
Zero Divisors

Let R be a commutative ring with identity. The zero-divisor graph of R, denoted Γ(R), is the simple graph whose vertices are the nonzero zero-divisors of R, and two distinct vertices x and y are linked by an edge if and only if xy=0. The genus of a simple graph G is the smallest integer g such that G can be embedded into an orientable surface Sg. In this paper, we determine that the genus of the zero-divisor graph of Zn, the ring of integers modulo n, is two or three.

scholarly journals Computing Vertex-Based Eccentric Topological Descriptors of Zero-Divisor Graph Associated with Commutative Rings

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Abdullah Ali H. Ahmadini ◽  
Ali N. A. Koam ◽  
Ali Ahmad ◽  
Martin Bača ◽  
Andrea Semaničová–Feňovčíková
Keyword(s):  
Commutative Ring ◽  
Communication Theory ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Topological Indices ◽  
Topological Descriptors ◽  
Zero Divisor Graph ◽  
Finite Commutative Ring

The applications of finite commutative ring are useful substances in robotics and programmed geometric, communication theory, and cryptography. In this paper, we study the vertex-based eccentric topological indices of a zero-divisor graphs of commutative ring ℤp2×ℤq, where p and q are primes.

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