Complemented zero-divisor graphs and Boolean rings

The zero-divisor graph of a commutative ring R, denoted by Γ(R), is a graph whose two distinct vertices x and y are joined by an edge if and only if xy = 0 or yx = 0. The main problem of the study of graphs defined on algebraic structure is to recognize finite rings through the properties of various graphs defined on it. The main objective of this article is to study the Wiener index of zero-divisor graph and compressed zero-divisor graph of the ring of integer modulo psqt for all distinct primes p, q and . We study the structure of these graphs by dividing the vertex set. Furthermore, a formula for the Wiener index of zero-divisor graph of Γ(R), and a formula for the Wiener index of associated compressed zero-divisor graph ΓE(R) are derived for .

The equivalence class [r] of an element r ∈ R is the set of zero-divisors s such that ann(r) = ann(s), that is, [r] = {s ∈ R : ann(r) = ann(s). The compressed zero-divisor graph, denoted by Γc(R), is the compression of a zero-divisor graph, in which the vertex set is the set of all equivalence classes of nonzero zero-divisors of a ring R, that is, the vertex set of Γc(R) is Re − {[0], [1]}, where Re = {[r] : r ∈ R} and two distinct equivalence classes [r] and [s] are adjacent if and only if rs = 0. In this article, we investigate the planarity of Γc(R) for some finite local rings of order p 2 , p 3 and determine the planarity of compressed zero-divisor graph of some local rings of order 32, whose zero-divisor graph is nonplanar. Further, we determine values of m and n for which Γc(Zn) and Γc(Zn[x]/(xm)) are planar.

The zero-divisor graph of a commutative ring R with unity is the graph Γ(R) whose vertex set is the set of nonzero zero divisors of R, where two vertices are adjacent if and only if their product in R is zero. A vertex coloring c : V (G) → ℕ of a non-trivial connected graph G is called a sigma coloring if σ(u) = σ(ν) for any pair of adjacent vertices u and v. Here, σ(χ) denotes the sum of the colors assigned to vertices adjacent to x. The sigma chromatic number of G, denoted by σ(G), is defined as the least number of colors needed to construct a sigma coloring of G. In this paper, we analyze the structure of the zero-divisor graph of rings ℤn, where n = pn1 1 P2 n2 …Pm nm, where m,ni,n2, …,nm are positive integers and p1,p2, …,pm are distinct primes. The analysis is carried out by partitioning the vertex set of such zero-divisor graphs and analyzing the adjacencies, cardinality, and the degree of the vertices in each set of the partition. Using these properties, we determine the sigma chromatic number of these zero-divisor graphs.

It was shown by C. Wickham [12] that “for a fixed positive integer g, there are finitely many isomorphism classes of finite commutative rings whose zero-divisor graph has genus g.” In this note, we give a short direct proof for this result. Moreover, we show that, if the zero-divisor graph of a commutative ring R has finite genus g, then either g = 0 or R is a finite ring. This immediately generalizes Wickham's theorem to arbitrary (not necessary finite) commutative rings.

Let [Formula: see text], [Formula: see text], [Formula: see text] denote the zero-divisor graph, compressed zero-divisor graph and annihilating ideal graph of a commutative ring [Formula: see text], respectively. In this paper, we prove that [Formula: see text] for a semisimple commutative ring [Formula: see text] and represent [Formula: see text] as a generalized join of a finite set of graphs. Further, we study the zero-divisor graph of a semisimple group-ring [Formula: see text] and proved several structural properties of [Formula: see text] and [Formula: see text], where [Formula: see text] is a field with [Formula: see text] elements and [Formula: see text] is a cyclic group with [Formula: see text] elements.

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.

On zero-divisor graphs of small finite commutative rings

ABSTRACTA graph Γ is said to be End-regular if its endomorphism monoid End(Γ) is regular. D. Lu and T. Wu [25] posed an open problem: Given a ring R, when does the zero-divisor graph Γ(R) have a regular endomorphism monoid? and they solved the problem for R a commutative ring with at least one nontrivial idempotent. In this paper, we solve this problem for zero-divisor graphs of group rings.

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.

This paper resolves the previously open problem of determining the metric dimension of the zero-divisor graph $\Gamma (R)$ for the Boolean ring $R = (\mathbb {Z}_2)^n$ . The unique structure of this graph, characterised by its diameter of $3$ and lack of common neighbours, has hindered all standard approaches. We introduce a novel combinatorial method that constructs an explicit resolving set. Consequently, we provide a precise formula for $\dim _m(\Gamma ((\mathbb {Z}_2)^n))$ , closing a notable gap in the literature on metric dimensions of zero-divisor graphs. As an application, we compute the metric dimension of a zero-divisor graph of a ring with a Boolean factor.

Let $R$ be a finite commutative ring with identity. We form the zero divisor graph of $R$ by taking the nonzero zero divisors as the vertices and connecting two vertices, $x$ and $y$, by an edge if and only if $xy=0$. We establish that if the zero divisor graph of a finite commutative ring with identity is planar, then the graph has a planar supergraph with a Hamiltonian cycle. We also determine the book thickness of all planar zero divisor graphs.

Let R be a commutative ring with identity and A(R) be the set of ideals with nonzero annihilator. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R)* = A(R)\{0} and two distinct vertices I and J are adjacent if and only if IJ = 0. In this paper, we study the domination number of AG(R) and some connections between the domination numbers of annihilating-ideal graphs and zero-divisor graphs are given.

In this paper, we characterize the perfect zero-divisor graphs and chordal zero-divisor graphs (its complement) of ordered sets. These results are applied to the zero-divisor graphs of finite reduced rings, the comaximal ideal graphs of rings, the annihilating ideal graphs of rings, the intersection graphs of ideals of rings, and the intersection graphs of subgroups of groups. In fact, it is shown that these graphs associated with a commutative ring R with identity can be effectively studied via the zero-divisor graph of a specially constructed poset from R.

In this paper, we introduce a new graph whose vertices are the non-zero zero-divisors of a commutative ring R, and for distincts elements x and y in the set Z(R)* of the non-zero zero-divisors of R, x and y are adjacent if and only if xy = 0 or x + y ∈ Z(R). We present some properties and examples of this graph, and we study its relationship with the zero-divisor graph and with a subgraph of the total graph of a commutative ring.

In this paper, we will extend the notion of zero-divisor graph of commutative rings to zero-divisor graph of abelian groups and study this zero-divisor graph. We characterize the zero-divisor graph of almost all abelian groups. Just a few classes of reduced abelian groups remain untouched in this paper.