Ring Zn Research Articles

Degree-based topological indices of the idempotent graph of the ring [formula omitted]

Let R be a finite commutative ring with a non-zero identity, and Id(R) be the set of idempotent elements of R. The idempotent graph of R, denoted by GId(R), is a simple undirected graph with all elements of R as vertices, and two distinct vertices u, v are adjacent if and only if u+v∈Id(R). In this paper, we consider the idempotent graph of the ring Zn and investigate some degree-based topological indices, such as the general sum-connectivity index, the general Randić index, the general Zagreb index, and the Sombor index of that graph by considering the M-polynomial of the graph.

Linear strand of edge ideals of zero divisor graphs of the ring ℤn

For a simple graph G with edge ideal I(G), we study the N -graded Betti numbers in the linear strand of the minimal free resolution of I ( Γ ( Z n ) ) , where Γ ( Z n ) is the zero divisor graph of the ring Z n . We present sharp bounds for the Betti numbers of Γ ( Z n ) and characterize the graphs attaining these bounds, thereby establishing the correct equality case for one of the results of the earlier published paper (Theorem 4.5, S. Pirzada and S. Ahmad, On the linear strand of edge ideals of some zero divisor graphs, Commun. Algebra 51(2) (2023) 620–632). Also, we present homological invariants of the edge rings of Γ ( Z n ) for n = p 2 q and pqr, with primes p < q < r .

Randić spectrum of the weakly zero-divisor graph of the ring ℤn

In this article, we find the Randić spectrum of the weakly zero-divisor graph of a finite commutative ring R with identity 1 ≠ 0 , denoted as W Γ ( R ) , where R is taken as the ring of integers modulo n . The weakly zero-divisor graph of the ring R is a simple undirected graph with vertices representing non-zero zero-divisors in R . Two vertices, denoted as a and b, are connected if there are elements x in the annihilator of a and y in the annihilator of b such that their product xy equals zero. In particular, we examine the Randić spectrum of W Γ ( Z n ) for specific values of n , which are products of prime numbers and their powers.

Properties of the ideal-intersection graph of the ring Zn

In this paper we study properties of the ideal-intersection graph of the ring Zn. The graph of ideal intersections is a simple graph in which the vertices are non-zero ideals of the ring, and two vertices (ideals) are adjacent if their intersection is also a non-zero ideal of the ring. These graphs can be referred to as the intersection scheme of equivalence classes (See: Laxman Saha, Mithun Basak Kalishankar Tiwary “Metric dimension of ideal-intersection graph of the ring Zn” [1] ).In this article we prove that the triameter of graph is equal to six or less than six. We also describe maximal clique of the ideal-intersection graph of the ring Zn. We prove that the chromatic number of this graph is equal to the sum of the number of elements in the zero equivalence class and the class with the largest number of element. In addition, we demonstrate that eccentricity is equal to 1 or it is equal to 2. And in the end we describe the central vertices in the ideal-intersection graph of the ring Zn.

On Normalized Laplacian Spectra of the Weakly Zero-Divisor Graph of the Ring ℤn

For a finite commutative ring R with identity 1≠0, the weakly zero-divisor graph of R denoted as WΓ(R) is a simple undirected graph having vertex set as a set of non-zero zero-divisors of R and two distinct vertices a and b are adjacent if and only if there exist elements r∈ann(a) and s∈ann(b) satisfying the condition rs=0. The zero-divisor graph of a ring is a spanning sub-graph of the weakly zero-divisor graph. This article finds the normalized Laplacian spectra of the weakly zero-divisor graph WΓ(R). Specifically, the investigation is carried out on the weakly zero-divisor graph WΓ(Zn) for various values of n.

On Sharp Bounds of Local Fractional Metric Dimension for Certain Symmetrical Algebraic Structure Graphs

The smallest set of vertices needed to differentiate or categorize every other vertex in a graph is referred to as the graph’s metric dimension. Finding the class of graphs for a particular given metric dimension is an NP-hard problem. This concept has applications in many different domains, including graph theory, network architecture, and facility location problems. A graph G with order n is known as a Toeplitz graph over the subset S of consecutive collections of integers from one to n, and two vertices will be adjacent to each other if their absolute difference is a member of S. A graph G(Zn) is called a zero-divisor graph over the zero divisors of a commutative ring Zn, in which two vertices will be adjacent to each other if their product will leave the remainder zero under modulo n. Since the local fractional metric dimension problem is NP-hard, it is computationally difficult to identify an optimal solution or to precisely determine the minimal size of a local resolving set; in the worst case, the process takes exponential time. Different upper bound sequences of local fractional metric dimension are suggested in this article, along with a comparison analysis for certain families of Toeplitz and zero-divisor graphs. Furthermore, we note that the analyzed local fractional metric dimension upper bounds fall into three metric families: constant, limited, and unbounded.

Spectrum of the Cozero-Divisor Graph Associated to Ring Zn

Let R be a commutative ring with identity 1≠0 and let Z(R)′ be the set of all non-unit and non-zero elements of ring R. Γ′(R) denotes the cozero-divisor graph of R and is an undirected graph with vertex set Z(R)′, w∉zR, and z∉wR if and only if two distinct vertices w and z are adjacent, where qR is the ideal generated by the element q in R. In this article, we investigate the signless Laplacian eigenvalues of the graphs Γ′(Zn). We also show that the cozero-divisor graph Γ′(Zp1p2) is a signless Laplacian integral.

Algebraic Structure Graphs over the Commutative Ring Zm: Exploring Topological Indices and Entropies Using M-Polynomials

The field of mathematics that studies the relationship between algebraic structures and graphs is known as algebraic graph theory. It incorporates concepts from graph theory, which examines the characteristics and topology of graphs, with those from abstract algebra, which deals with algebraic structures such as groups, rings, and fields. If the vertex set of a graph G^ is fully made up of the zero divisors of the modular ring Zn, the graph is said to be a zero-divisor graph. If the products of two vertices are equal to zero under (modn), they are regarded as neighbors. Entropy, a notion taken from information theory and used in graph theory, measures the degree of uncertainty or unpredictability associated with a graph or its constituent elements. Entropy measurements may be used to calculate the structural complexity and information complexity of graphs. The first, second and second modified Zagrebs, general and inverse general Randics, third and fifth symmetric divisions, harmonic and inverse sum indices, and forgotten topological indices are a few topological indices that are examined in this article for particular families of zero-divisor graphs. A numerical and graphical comparison of computed topological indices over a proposed structure has been studied. Furthermore, different kinds of entropies, such as the first, second, and third redefined Zagreb, are also investigated for a number of families of zero-divisor graphs.

DOMINATION PARAMETERS ON BIPARTITE GRAPHS OF THE COMMUTATIVE RING ZN

In 1977,E.J.Cockayne and S.T. Hedetniemi conducted a commendable and broad survey on the outcomes of the existing concepts of dominating set in graphs. It’s basic concept is the dominating set and the domination number. In this paper our main aim is to find out the dominating parameters of Bipartite graph obtained on the Commutative ring of type Zn. A subset D of V is said to be a dominating set of G if every vertex in V-D is adjacent to a vertex in D. A bipartite graph is a graph G whose vertex set is partitioned into two disjoint subsets X and Y such that each edge in G has one end in X and the other end in Y. We determine the domination parameters i.e., domination number, dominating set and Minimum domination number of Bipartite graphs of the commutative ring Zn.

On Monochromatic Clean Condition on Certain Finite Rings

For a finite commutative ring R, let a,b,c∈R be fixed elements. Consider the equation ax+by=cz where x, y, and z are idempotents, units, and any element in the ring R, respectively. We say that R satisfies the r-monochromatic clean condition if, for any r-colouring χ of the elements of the ring R, there exist x,y,z∈R with χ(x)=χ(y)=χ(z) such that the equation holds. We define m(a,b,c)(R) to be the least positive integer r such that R does not satisfy the r-monochromatic clean condition. This means that there exists χ(i)=χ(j) for some i,j∈{x,y,z} where i≠j. In this paper, we prove some results on m(a,b,c)(R) and then formulate various conditions on the ring R for when m(1,1,1)(R)=2 or 3, among other results concerning the ring Zn of integers modulo n.

Calculating different indices of Idempotent graph of ring ℤn

By Idempotent graph of ℤn we mean the graph whose vertices are all elements of ℤn such that distinct vertices u and v are adjacent if and only if u + v is an idempotent element of ℤn, denoted by Gid(ℤn). In this paper, we compute some kind of topological indices of Idempotent graph of ℤn.

Proposing an efficient implementation method for exponentiation in digital signature scheme on ring Zn

In this paper, we propose a design method for the signature scheme based on ring structure Zn. Our signature schemes are more secure, generate signatures at a faster rate than that of the ElGamal scheme and its variants. Moreover, our approaches also overcome some disadvantages of some similar signature schemes on ring Zn. For these advantages, they are fully applicable in practice.

Optimal Method of Integer Factorization

The object of the research is performance of integer factorization algorithms and possibility of mathematical methods using in these algorithms. The subject of the research is cryptographic properties GNFS method. Methods of research are methods of the theory of elliptic curves, finite fields, abstract algebra and advanced the theory of factorization algorithms. As a result of this work, the dependence of the properties of a minimal time of factorization and choice of algebraic factor bases over the ring Zn, where n=pq was established. Moreover, we have implemented the general number field sieve (GNFS), which is the most efficient classical algorithm known for factoring integers.

Non-Braid Graphs of Ring Zn

The research in graph theory has been widened by combining it with ring. In this paper, we introduce the definition of a non-braid graph of a ring. The non-braid graph of a ring R, denoted by YR, is a simple graph with a vertex set R\B(R), where B(R) is the set of x in R such that xyx=yxy for all y in R. Two distinct vertices x and y are adjacent if and only if xyx not equal to yxy. The method that we use to observe the non-braid graphs of Zn is by seeing the adjacency of the vertices and its braider. The main objective of this paper is to prove the completeness and connectedness of the non-braid graph of ring Zn. We prove that if n is a prime number, the non-braid graph of Zn is a complete graph. For all n greater than equal to 3, the non-braid graph of Zn is a connected graph.

Pancyclic zero divisor graph over the ring ℤn[i

Let [Formula: see text] be the zero divisor graph over the ring [Formula: see text]. In this paper, we study pancyclic properties of [Formula: see text] and [Formula: see text], respectively, for different [Formula: see text]. Also, we prove some results in which [Formula: see text] and [Formula: see text] are pancyclic for different values of [Formula: see text].

Comments on the Clique Number of Zero‐Divisor Graphs of Zn

In 2008, J. Skowronek‐kaziw extended the study of the clique number ω(G(Zn)) to the zero‐divisor graph of the ring Zn, but their result was imperfect. In this paper, we reconsider ω(G(Zn)) of the ring Zn and give some counterexamples. We propose a constructive method for calculating ω(G(Zn)) and give an algorithm for calculating the clique number of zero‐divisor graph. Furthermore, we consider the case of the ternary zero‐divisor and give the generation algorithm of the ternary zero‐divisor graphs.

An association between digraphs and rings

In the present article, we are going to highlight the relation between different digraphs (cycles) of finite commutative ring Zn for a natural number n, under the map (a,b) ? (a+b,ab). The algorithm, which is used to perform the calculations, has been built in MATLAB?.

On normalized Laplacian spectrum of zero divisor graphs of commutative ring ℤn

For a finite commutative ring ℤ n with identity 1 ≠ 0 , the zero divisor graph Γ (ℤ n ) is a simple connected graph having vertex set as the set of non-zero zero divisors, where two vertices x and y are adjacent if and only if xy=0. We find the normalized Laplacian spectrum of the zero divisor graphs Γ (ℤ n ) for various values of n and characterize n for which Γ (ℤ n ) is normalized Laplacian integral. We also obtain bounds for the sum of graph invariant S β * ( G ) -the sum of the β -th power of the non-zero normalized Laplacian eigenvalues of Γ (ℤ n ) .

Perfect Codes Over Induced Subgraphs of Unit Graphs of Ring of Integers Modulo n

The induced subgraph of a unit graph with vertex set as the idempotent elements of a ring R is a graph which is obtained by deleting all non idempotent elements of R. Let C be a subset of the vertex set in a graph Γ. Then C is called a perfect code if for any x, y ∈ C the union of the closed neighbourhoods of x and y gives the the vertex set and the intersection of the closed neighbourhoods of x and y gives the empty set. In this paper, the perfect codes in induced subgraphs of the unit graphs associated with the ring of integer modulo n, Zn that has the vertex set as idempotent elements of Zn are determined. The rings of integer modulo n are classified according to their induced subgraphs of the unit graphs that accept a subset of a ring Zn of different sizes as the perfect codes

On the adjacency spectrum of zero divisor graph of ring ℤn

The zero divisor graph [Formula: see text] of a commutative ring [Formula: see text] with unity is a simple undirected graph whose vertices are 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 graphical structure and the adjacency spectrum of the zero divisor graph of ring [Formula: see text]. For any non-prime positive integer [Formula: see text] with [Formula: see text] number of proper divisors, we show that the adjacency spectrum of [Formula: see text] consists of the eigenvalues of a symmetric matrix [Formula: see text] of size [Formula: see text], and at the most [Formula: see text] and [Formula: see text]. Also, we find the exact multiplicity of the eigenvalue [Formula: see text] and show that all eigenvalues of [Formula: see text] are nonzero, by determining the rank and nullity of the adjacency matrix of [Formula: see text]. We find the values of [Formula: see text] for which the adjacency spectrum of [Formula: see text] contains only nonzero eigenvalues. Finally, by computing the characteristic polynomial of the matrix [Formula: see text], we determine the characteristic polynomial of [Formula: see text] whenever [Formula: see text] is a prime power.