Adjacency matrix and eigenvalues of the zero divisor graph Γ(Zn)

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 AG([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.

Let be a commutative ring with , and be an -module. The minimal intersection graph of , denoted by is a simple undirected graph whose vertices are proper non-zero submodules of and any two distinct vertices and are adjacent if and only if be an minimal (= simple) submodule of . In this article, we explore connectedness, clique number, split character, planarity, independence number and domination number of .

Let [Formula: see text] be a ring with unity. The cozero-divisor graph of a ring [Formula: see text] is an undirected simple graph whose vertices are the set of all nonzero and non-unit elements of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. The reduced cozero-divisor graph of a ring [Formula: see text] is an undirected simple graph whose vertex set is the set of all nontrivial principal ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. Jesili et al. [On the genus of reduced cozero-divisor graph of commutative rings, Soft Comput. 27(2) (2023) 657–666] classified all non-local commutative rings whose reduced cozero-divisor graph is of genus one. In this paper, we characterize all classes of finite non-local commutative rings whose reduced cozero-divisor graph is of genus two. Moreover, we classify all the non-local commutative rings whose cozero-divisor graph is of genus two.

Let R be a ring and , where and are the sets of all left and right zero-divisors of R, respectively. The zero-divisor graph of R, denoted by , is a simple undirected graph with vertex set , and two distinct vertices are adjacent if and only if ab = 0 or ba = 0. Let , the ring of integers modulo n, and the ring of all matrices over a finite field of q elements. In this article, using the technique on characteristic matrices, we give the value of fixing number of . Moreover, we calculate the fixing number and metric dimension of and .

<abstract><p>The aim of this paper is to introduce and study the concept of a hyperideal-based zero-divisor graph associated with a general hyperring. This is a generalized version of the zero-divisor graph associated with a commutative ring. For any general hyperring $ R $ having a hyperideal $ I $, the $ I $-based zero-divisor graph $ \Gamma^{(I)}(R) $ associated with $ R $ is the simple graph whose vertices are the elements of $ R\setminus I $ having their hyperproduct in $ I $, and two distinct vertices are joined by an edge when their hyperproduct has a non-empty intersection with $ I $. In the first part of the paper, we concentrate on some general properties of this graph related to absorbing elements, while the second part is dedicated to the study of the $ I $-based zero-divisor graph associated to the general hyperring $ \mathbb{Z}_n $ of the integers modulo $ n $, when $ n = 2p^mq $, with $ p $ and $ q $ two different odd primes, and fixing the hyperideal $ I $.</p></abstract>

Let G be a finite abelian group with identity 0. For an integer the additive power graph of G is the simple undirected graph with vertex set G in which two distinct vertices x and y are adjacent if and only if x + y = nt for some with When the additive power graph has been studied in the name of square graph of finite abelian groups. In this paper, we study the additive power graph of G with n = 3 and name the graph as the cubic power graph. The cubic power graph of G is denoted by More specifically, we obtain the diameter and the girth of the graph and its complement Using these, we obtain a condition for and its complement to be self-centered. Also, we obtain the independence number, the clique number and the chromatic number of and its complement and hence we prove that and its complement are weakly perfect. Also, we discuss about the perfectness of At last, we obtain a condition for and its complement to be vertex pancyclic.

For a graph G(V, E) with order n ≥ 2, the locating code of a vertex v is a finite vector representing distances of v with respect to vertices of some ordered subset W of V(G). The set W is a locating set of G(V, E) if distinct vertices have distinct codes. A locating set containing a minimum number of vertices is a minimum locating set for G(V, E). The locating number denoted by loc (G) is the number of vertices in the minimum locating set. Let R be a commutative ring with identity 1 ≠ 0, the zero-divisor graph denoted by Γ(R), is the (undirected) graph whose vertices are the nonzero zero-divisors of R with two distinct vertices joined by an edge when the product of vertices is zero. We introduce and investigate locating numbers in zero-divisor graphs of a commutative ring R. We then extend our definition to study and characterize the locating numbers of an ideal based zero-divisor graph of a commutative ring R.

Let [Formula: see text] be a finite commutative ring and [Formula: see text] is nilpotent for some [Formula: see text]. The nilpotent graph [Formula: see text] of [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] in which two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is nilpotent. In this paper, we observe a relationship between zerodivisor graphs, essential graphs and nilpotent graphs of [Formula: see text]. In continuation of genus characterizations in [T. Asir K. Mano and T. Tamizh Chelvam, Correction to: Classification of non-local rings with genus two zerodivisor graphs, Soft Comput. 25 (2021) 3355–3356; K. Selvakumar and M. Subajini, Finite commutative ring with genus two essential 10 graph, J. Algebra Appl. 16(2) (2018) 1850121], we classify all finite commutative rings (up to isomorphism) whose nilpotent graphs are of genus two.

Let R be a commutative ring with unity, the extended zero divisor graph E(R) is defined by considering the non-zero zero divisors Z(R)∗ as the vertices and two distinct vertices x and y of R are adjacent whenever there exist two positive integers m and n such that xmym=0 with xm≠0 and yn≠ 0. In this paper, we investigate metric dimension, fault tolerant metric dimension and local metric dimension of the extended zero divisor graph for the ring of integers modulo m and the ring of Gaussian integers modulo m.

لتكن حلقة ابدالية أحادية، وليكن مقاساً ايسر احادي. بيان تقاطع صغير من النمط-g للمقاس يرمز له , هو بيان غير مباشر بسيط رؤوسه تقابل كل المقاسات الجزئية غير التافهة في وكل رأسين مختلفين متجاورين أذا وفقط أذا كان التقاطع بينهما هو صغير من النمط-g. في هذا المقال، تم دراسة العلاقة بين الخصائص الجبرية الى والخصائص البيانية الى . أعتبر خصائص مثل الترابط، الكمال. علاوة على ذلك، يتم تحدد القطر و الطوق الى ، وكذلك أعطى صيغة لحساب الزمرة الى جانب أعداد الهيمنة الى .

Let R R be a commutative finite semiring with nonzero identity and H H be an arbitrary multiplicatively closed subset R R . The generalized identity-summand graph of R R is the (simple) graph G H ( R ) GH(R) with all elements of R R as the vertices, and two distinct vertices x x and y y are adjacent if and only if x + y ∈ H x+y∈H . In this paper, we study some basic properties of G H ( R ) GH(R) . Moreover, we characterize the planarity, chromatic number, clique number and independence number of G H ( R ) GH(R) .

The zero-divisor graph Γ(R) of an associative ring R is the graph whose vertices are all nonzero zero-divisors (one-sided and two-sided) of R, and two distinct vertices x and y are joined by an edge if and only if either xy = 0 or yx = 0. In the present paper, we study some properties of ring varieties where every finite ring is uniquely determined by its zero-divisor graph.

For any nonempty set Ω and k-subset Λ, the k-intersection graph, denoted by Γm(Ω,Λ), is an undirected simple graph whose vertices are all m-subsets of Ω and two distinct vertices A and B are adjacent if and only if A∩B ⊈ Λ. In this paper, we determine diameter, girth, some numerical invariants and planarity, Hamiltonian and perfect matching of these graphs. finally we investigate their adjacency matrices.

On the coloring of the annihilating-ideal graph of a commutative ring

On the genus of nil-graph of ideals of commutative rings