On the Unit Graph of a Noncommutative Ring

Algebras and Representation Theory

Journal of Algebra and Its Applications, 2012

Let R be a commutative ring with nonzero identity. For an arbitrary multiplicatively closed subset S of R, we associate a simple graph denoted by ΓS(R) with all elements of R as vertices, and two distinct vertices x, y ∈ R are adjacent if and only if x+y ∈ S. Two well-known graphs of this type are the total graph and the unit graph. In this paper, we study some basic properties of ΓS(R). Moreover, we will improve and generalize some results for the total and the unit graphs.

2021

Let R be a commutative ring with 1. In [3], we introduced a graph G(R) whose vertices are elements of R and two distinct vertices a, b are adjacent if and only if aR + bR = eR for some non-zero idempotent e in R. Let G′(R) be the subgraph of G(R) generated by the non-units of R. In this paper, we characterize those rings R for which the graph G′(R) is connected and Eulerian. Also we characterize those rings R for which genus of the graph G′(R) is ≤ 2. Finally, we show that the graph G′(R) is a line graph of some graph if and only if R is either a regular ring or a local ring.AMS Subject Classification 2020 : 05C25

2013

Let R be a commutative ring with nonzero identity and H be a nonempty proper subset of R such that R\H is a saturated multiplicatively closed subset of R. The generalized total graph of R is the (simple) graph GT H (R) with all elements of R as the vertices, and two distinct vertices x and y are adjacent if and only if x + y ∈ H. In this paper, we investigate the structure of GT H (R).

Communications in Algebra, 2010

Let R be a ring with nonzero identity. The unit graph of R, denoted by G R , has its set of vertices equal to the set of all elements of R; distinct vertices x and y are adjacent if and only if x + y is a unit of R. In this article, the basic properties of G R are investigated and some characterization results regarding connectedness, chromatic index, diameter, girth, and planarity of G R are given. (These terms are defined in Definitions and Remarks 4

Axioms

Let R be a finite ring and r∈R. The r-noncommuting graph of R, denoted by ΓRr, is a simple undirected graph whose vertex set is R and two vertices x and y are adjacent if and only if [x,y]≠r and [x,y]≠−r. In this paper, we obtain expressions for vertex degrees and show that ΓRr is neither a regular graph nor a lollipop graph if R is noncommutative. We characterize finite noncommutative rings such that ΓRr is a tree, in particular a star graph. It is also shown that ΓR1r and ΓR2ψ(r) are isomorphic if R1 and R2 are two isoclinic rings with isoclinism (ϕ,ψ). Further, we consider the induced subgraph ΔRr of ΓRr (induced by the non-central elements of R) and obtain results on clique number and diameter of ΔRr along with certain characterizations of finite noncommutative rings such that ΔRr is n-regular for some positive integer n. As applications of our results, we characterize certain finite noncommutative rings such that their noncommuting graphs are n-regular for n≤6.

2019

Let R be a finite ring and r ∈ R. The r-noncommuting graph of R, denoted by ΓrR, is a simple undirected graph whose vertex set is R and two vertices x and y are adjacent if and only if [x, y] 6= r and −r. In this paper, we study several properties of ΓrR. We show that Γ r R is not a regular graph, a lollipop graph and complete bipartite graph. Further, we consider an induced subgraph of ΓrR (induced by the non-central elements of R) and obtained some characterizations of R.

Linear Algebra and Its Applications, 2009

Let R be a ring (not necessarily commutative) with 1. Following Sharma and Bhatwadekar [P.K. Sharma, S.M. Bhatwadekar, A note on graphical representation of rings, J. Algebra 176 (1995) 124-127], we define a graph on R, (R), with vertices as elements of non-units of R, where two distinct vertices a and b are adjacent if and only if Ra + Rb = R. In this paper, we investigate the behavior of (R). We are able to prove that if R is left Artinian then (R) − J(R) is connected and if (R) − J(R) is a forest then (R) − J(R) is a star graph, where J(R) is the Jacobson radical of R. For any finite field F q , we obtain the minimal degree, the maximal degree, the connectivity, the clique number and the chromatic number of (M n (F q )). Finally, for any finite field and any integer n 2, we prove that if R is a ring with identity and (R) ∼ = (M n (F)), then R ∼ = M n (F). We also prove that if R and S are two finite commutative rings with identity, and n, m 2 such that (M n (R)) ∼ = (M m (S)), then n = m and R ∼ = S provided that R is reduced.

Communications in Algebra, 2013

2014

Let R be a (commutative) ring with nonzero identity and Z.R/ be the set of all zero divisors of R. The total graph of R is the simple undirected graph T. .R// with vertices all elements of R, and two distinct vertices x and y are adjacent if and only if x C y 2 Z.R/. This type of graphs has been studied by many authors. In this paper, we state many of the main results on the total graph of a ring and its related graphs.