Graph Of Ring Research Articles

Graphs from matrices - a survey

Let R be a commutative ring with identity. For a positive integer n ≥ 2 , let M n ( R ) be the set of all n × n matrices over R and M n ( R ) * be the set of all non-zero matrices of M n ( R ) . The zero-divisor graph of M n ( R ) is a simple directed graph with vertex set the non-zero zero-divisors in M n ( R ) and two distinct matrices A and B are adjacent if their product is zero. Given a matrix A ∈ M n ( R ) , Tr(A) is the trace of the matrix A. The trace graph of the matrix ring M n ( R ) , denoted by Γ t ( M n ( R ) ) , is the simple undirected graph with vertex set { A ∈ M n ( R ) * : there exists B ∈ M n ( R ) * such that Tr ( AB ) = 0 } and two distinct vertices A and B are adjacent if and only if Tr ( AB ) = 0. For an ideal I of R, the notion of the ideal based trace graph, denoted by Γ I t ( M n ( R ) ) , is a simple undirected graph with vertex set M n ( R ) ∖ M n ( I ) and two distinct vertices A and B are adjacent if and only if Tr ( AB ) ∈ I . In this survey, we present several results concerning the zero-divisor graph, trace graph and the ideal based trace graph of matrices over R.

On L(2, 1)-labeling of zero-divisor graphs of finite commutative rings

On L(2, 1)-labeling of zero-divisor graphs of finite commutative rings

On the line graph structure of the cozero-divisor graph of a commutative ring

On the line graph structure of the cozero-divisor graph of a commutative ring

CONSTRUCT THE TRIPLE ZERO GRAPH OF RING Z_n USING PYTHON

Let be a commutative ring with nonzero identity and there exists such that , , , denotes the set of all triple zero elements of . The triple zero graph of , denoted by , is an undirected graph with vertex set where two distinct vertices and are adjacent if and only if , and there exists a nonzero element of such that , , and . Python is a programming language with simple and easy-to-learn code that can be used to solve problems in algebra and graphs. In this paper, we construct the triple zero graph of ring using Python. Based on the output of the program, several properties of are obtained, such as if and , then is a planar graph, if with is prime numbers, then is a complete graph , and if with is prime numbers and , then is a connected graph.

On the comaximal graph of a ring

In this paper, we prove that the study of the comaximal graph of a finite commutative ring with identity is the same as the study of the zero-divisor graph of the specially constructed lattice. Also, we introduce the blow-up of a Boolean lattice using finite chains. Furthermore, we prove that in the case of a reduced ring, the comaximal graph and the co-annihilating graph coincide. In the last section, we study the perfectness of the comaximal graphs and characterize the Hamiltonian property of the complement of a comaximal graph.

Some properties of generalized comaximal graph of commutative ring

Some properties of generalized comaximal graph of commutative ring

Topological Indices and Properties of the Prime Ideal Graph of a Commutative Ring and Its Line Graph

Let R be a commutative ring with identity, and P be a prime ideal of R. The prime ideal graph, denoted by Γp, is the graph where the set of vertices is R/{0} and two vertices are joined by an edge if their product belongs to P. This paper will discuss topological indices and some properties of the prime ideal graph of a commutative ring and its line graph. Topological indices, such as the Wiener, first Zagreb, second Zagreb, Harary, Gutman, Schultz, and Harmonic indices, are related to the degree of vertices and the diameter of the graphs. In this paper, we also discuss the independence and domination numbers of the line graph.

Dominacijski broj za ciklooktanske lance

Introduction/purpose: Chemical structures are conveniently represented by graphs where atoms are nodes (vertices) and chemical bonds are branches (lines) in the graph. A graphical representation of a molecule provides a lot of useful information about the chemical properties of the molecule. It is known that numerous physical and chemical properties of molecules are highly correlated with theoretical invariants of graphs, which we call topological indices. One such theoretical invariant is the dominance number. The aim of this research is to determine the k-dominance number for cyclooctane chains 𝐶𝑂𝐶𝑛 1 , 𝐶𝑂𝐶𝑛 2 , 𝐶𝑂𝐶𝑛 3 and 𝐶𝑂𝐶𝑛 4 , for k ∈ {1,2,3}, n ∈ 𝑁. Methods: The cyclooctane chain is a chain of octagons connected by a single line. The vertices of the octagon are treated as nodes of the graph, and the sides and the line connecting them, as branches in the graph. Using mathematical methods, k-dominance was determined on one octagon, k∈{1,2,3}. Then, by representing the cyclooctane chains 𝐶𝑂𝐶𝑛 1 , 𝐶𝑂𝐶𝑛 2 , 𝐶𝑂𝐶𝑛 3 and 𝐶𝑂𝐶𝑛 4 , in a convenient, isomorphic way, we determined their k-dominance number, k∈{ 1,2,3}. Results: Determining k-dominance, k∈{1,2,3}, for 4 cyclooctane chains 𝐶𝑂𝐶𝑛 1 , 𝐶𝑂𝐶𝑛 2 , 𝐶𝑂𝐶𝑛 3 and 𝐶𝑂𝐶𝑛 4 , we obtained 12 different formulas to calculate their k-dominance number. All formulas are composed of several alternative algebraic expressions, the selection of which is conditioned by the divisibility of the number n by the number 2, 3 or 4, depending on the type of cyclooctane chain and k-dominance to be determined. The results of the research are fully presented in the paper through mathematically proven theorems and graphical representations. Conclusion: The results show that the k-dominance numbers, k∈{1,2,3}, on cyclooctane chains 𝐶𝑂𝐶𝑛 1 , 𝐶𝑂𝐶𝑛 2 , 𝐶𝑂𝐶𝑛 3 and 𝐶𝑂𝐶𝑛 4 , are determined and explicitly expressed by mathematical expressions. They also indicate the possibility of their application in molecular graphs of cyclooctane rings, in computational chemistry, chemical and biological industry.

Some remarks on the dominating sets of the annihilating-ideal graph of a commutative ring

Some remarks on the dominating sets of the annihilating-ideal graph of a commutative ring

Szeged Index and Padmakar-Ivan Index of Nilpotent Graph of Integer Modulo Ring with Prime Power Order

Recently, graphs have started to be used to represent a finite ring. Nikmehr and Khojasteh in the article defined the nilpotent graph of a ring . Denoted , is a graph with the set of vertices being all the elements in the ring Two vertices and are adjacent if and only if is nilpotent elements in the ring . Topological index is a field that discusses graph structure based on the degree of each vertex of a graph and the distance between vertices. In this study, the author will gives the general formula of the Szeged index and Padmakar-Ivan index of the nilpotent graph graph of the modulo ring with prime power order. The result of this research is a general formula for the topological indices of nilpotent graphs of the integer modulo ring, called the Szeged index and the Padmakar-Ivan index.

Connectedness in Projection Graphs of Commutative Rings

Connectedness in Projection Graphs of Commutative Rings

Metric dimension in a prime ideal sum graph of a commutative ring

The prime ideal sum graph of a commutative unital ring [Formula: see text], denoted by [Formula: see text], is an undirect and simple graph whose vertices are nontrivial ideals of [Formula: see text] and there exists an edge between two distinct vertices if and only if their sum is a prime ideal of [Formula: see text]. In this paper, the metric dimension of [Formula: see text] is discussed and some formulae for this parameter in [Formula: see text] are given.

Signless Laplacian spectrum of the cozero-divisor graph of the commutative ring ℤ𝑛

Abstract Let 𝑅 be a commutative ring with identity 1 ≠ 0 1\neq 0 and let Z ⁢ ( R ) ′ Z(R)^{\prime} be the set of all non-zero and non-unit elements of ring 𝑅. Further, Γ ′ ⁢ ( R ) \Gamma^{\prime}(R) denotes the cozero-divisor graph of 𝑅, is an undirected graph with vertex set Z ⁢ ( R ) ′ Z(R)^{\prime} , and w ∉ z ⁢ R w\notin zR and z ∉ w ⁢ R z\notin wR if and only if two distinct vertices 𝑤 and 𝑧 are adjacent, where q ⁢ R qR is the ideal generated by the element 𝑞 in 𝑅. In this paper, we find the signless Laplacian eigenvalues of the graphs Γ ′ ⁢ ( Z n ) \Gamma^{\prime}(\mathbb{Z}_{n}) for n = p 1 N ⁢ p 2 ⁢ p 3 n=p_{1}^{N}p_{2}p_{3} and p 1 N ⁢ p 2 M ⁢ p 3 p_{1}^{N}p_{2}^{M}p_{3} , where p 1 , p 2 , p 3 p_{1},p_{2},p_{3} are distinct primes and N , M N,M are positive integers. We also show that the cozero-divisor graph Γ ′ ⁢ ( Z p 1 ⁢ p 2 ) \Gamma^{\prime}(\mathbb{Z}_{p_{1}p_{2}}) is a signless Laplacian integral.

Gamma Graph Γ(Z_n)(γ) Of Zero Divisor Graph Of A Finite Commutative Ring Z_n

Consider the family of γ-sets of a zero-divisor graph of finite commutative ring and define the γgraphs (γ) = ( of to be the graph whose vertice V(γ) corresponds 1-to-1 with the γ-sets , say S1 and S2, form an edge in E(γ) if there exist a vertex vÎ such that (i)v is adjacent to and (ii) and Using this definition, we investigate the interplay between the graph theoretic properties of and and the ring theoretic properties of Further, we prove that are an Eulerian and Hamiltonian.

Some applications of eigenvalues of unitary Cayley graphs of matrix rings over finite fields

ABSTRACT In this work, we provide two applications of the eigenvalues of the unitary Cayley graphs over matrix rings over finite fields. For a ring R, denotes the Jacobson radical of R. In 2012, Kiani and Aghaei conjectured that if R and S are finite rings and their unitary Cayley graphs are isomorphic, then and are isomorphic. If this conjecture holds and , then R is characterized by its unitary Cayley graph, and we say that R is a ring determined by unitary Cayley graphs. Kiani and Aghaei showed that every finite commutative ring and are such rings. We examine the eigenvalues of and obtain many new families of rings determined by unitary Cayley graphs. In 2021, Podestá and Videla characterized all finite commutative rings R such that the graphs in the triple are mutually equienergetic non-isospectral and Ramanujan where is the unitary Cayley sum graph. We use the eigenvalues of the graph and some observation on the number of matrices of the given rank to extend Podestá and Videla's results by working on the finite non-commutative ring being a product of matrix rings over local rings.

The diameter of commuting graph of matrix ring of division rings

The commuting graph of a non-commutative ring [Formula: see text] denoted by [Formula: see text] is a graph whose vertices are non-central elements of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text]. Let [Formula: see text] be a division ring, [Formula: see text] and [Formula: see text]. In this paper, we show that if [Formula: see text] or [Formula: see text] and the field [Formula: see text] has no extension of degree [Formula: see text] then [Formula: see text] is a connected graph whose diameter is less than five. Also we prove that the diameter of [Formula: see text] is four where [Formula: see text] is a quaternion algebra over a field [Formula: see text] which has no extension of degree [Formula: see text] and [Formula: see text].

Coloring in essential annihilating-ideal graphs of commutative rings

Coloring in essential annihilating-ideal graphs of commutative rings

Some results on a supergraph of the sum annihilating ideal graph of a commutative ring

The rings considered in this paper are commutative with identity which are not integral domains. Let [Formula: see text] be a ring. An ideal [Formula: see text] of [Formula: see text] is said to be an annihilating ideal if there exists [Formula: see text] such that [Formula: see text]. Let [Formula: see text] denote the set of all annihilating ideals of [Formula: see text] and we denote [Formula: see text] by [Formula: see text]. With [Formula: see text], in this paper, we associate an undirected graph denoted by [Formula: see text] whose vertex set is [Formula: see text] and two distinct vertices [Formula: see text] are adjacent in this graph if and only if either [Formula: see text] or [Formula: see text]. The aim of this paper is to study the interplay between some graph properties of [Formula: see text] and the algebraic properties of [Formula: see text] and to compare some graph properties of [Formula: see text] with the corresponding graph properties of the annihilating ideal graph of [Formula: see text] and the sum annihilating ideal graph of [Formula: see text].

On the independence number of regular graphs of matrix rings

Consider a graph on the non-singular matrices over a finite field, in which two distinct non-singular matrices are joined by an edge whenever their sum is singular. We prove an upper bound for the independence number of this graph. As a consequence, we obtain a lower bound for its chromatic number that significantly improves a previous result of Tomon.

Regular divisor graph of finite commutative ring

Let R be a finite commutative ring with identity 1. We introduce a new graph called regular divisor graph and denoted by . We classify the finite commutative ring to get a special graph and we are going to study some properties of this graph, clique number, chromatic number, number of cycles, connectivity and blocks.