Related papers: On the Clean Graph of a Ring

The commuting graph of a non-commutative ring $R$ with center $Z(R)$ is a simple undirected graph whose vertex set is $R\setminus Z(R)$ and two vertices $x, y$ are adjacent if and only if $xy = yx$. In this paper, we compute the spectrum…

A ring R is a strongly 2-nil-clean if every element in R is the sum of two idempotents and a nilpotent that commute. A ring R is feebly clean if every element in R is the sum of two orthogonal idempotents and a unit. In this paper, strongly…

The upper ideal relation graph $\Gamma_{U}(R)$ of a commutative ring $R$ with unity is a simple undirected graph with the set of all non-unit elements of $R$ as a vertex set and two vertices $x$, $y$ are adjacent if and only if the…

A ring is clean (resp. almost clean) if each of its elements is the sum of a unit (resp. regular element) and an idempotent. In this paper we define the analogous notion for *-rings: a *-ring is *-clean (resp. almost *-clean) if its every…

A unital ring is called clean (resp. strongly clean) if every element can be written as the sum of an invertible element and an idempotent (resp. an invertible element and an idempotent that commutes). T.Y. Lam proposed a question: which…

A ring $R$ is (strongly) 2-nil-clean if every element in $R$ is the sum of two idempotents and a nilpotent (that commute). Fundamental properties of such rings are discussed. Let $R$ be a 2-primal ring. If $R$ is strongly 2-nil-clean, we…

The Wiener index of a graph $W(G)$ is a well studied topological index for graphs. An outstanding problem of \v{S}olt{\'e}s is to find graphs $G$ such that $W(G)=W(G-v)$ for all vertices $v\in V(G)$, with the only known example being…

The unitary Cayley graph $C_R$ of a finite unital ring $R$ is the simple graph with vertex set $R$ in which two elements $x$ and $y$ are connected by an edge if and only if $x-y$ is a unit of $R$. We characterize the unitary Cayley graph…

The essential annihilating-ideal graph $\mathcal{EG}(R)$ of a commutative unital ring $R$ is a simple graph whose vertices are non-zero ideals of $R$ with non-zero annihilator and there exists an edge between two distinct vertices $I,J$ if…

Motivated by the concept of clean index of rings, we introduce the concept of weak clean index of rings. For any element $a$ of a ring $R$ with unity, we define $ \chi(a)=\{e\in R\mid e^2=e\text{ and }a-e \mbox{ or } a+e \mbox{ is a…

Let $R$ be a ring with unity. The cozero-divisor graph of a ring $R$ is an undirected simple graph whose vertices are the set of all non-zero and non-unit elements of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $x…

Fullerene graphs are mathematical models of fullerene molecules. The Wiener $(r,s)$-complexity of a fullerene graph $G$ with vertex set $V(G)$ is the number of pairwise distinct values of $(r,s)$-transmission $tr_{r,s}(v)$ of its vertices…

A graph is called weakly perfect if its vertex chromatic number equals its clique number. Let $R$ be a ring and $I(R)^*$ be the set of all left proper non-trivial ideals of $R$. The intersection graph of ideals of $R$, denoted by $G(R)$, is…

An element of a ring is unique clean if it can be uniquely written as the sum of an idempotent and a unit. A ring $R$ is uniquely $\pi$-clean if some power of every element in $R$ is uniquely clean. In this article, we prove that a ring $R$…

Given a simple connected undirected graph G, the Wiener index W(G) of G is defined as half the sum of the distances over all pairs of vertices of G. In practice, G corresponds to what is known as the molecular graph of an organic compound.…

A ring $R$ is said to be clean if each element of $R$ can be written as the sum of a unit and an idempotent. In a recent article (J. Algebra, 405 (2014), 168-178), Immormino and McGoven characterized when the group ring $\mathbb…

The Wiener index $W(G)$ of a connected graph $G$ is a sum of distances between all pairs of vertices of $G$. In 1991, \v{S}olt\'{e}s formulated the problem of finding all graphs $G$ such that for every vertex $v$ the equation $W(G)=W(G-v)$…

Let $W(G)$ be the Wiener index of a graph $G$. We say that a vertex $v \in V(G)$ is a \v{S}olt\'es vertex in $G$ if $W(G - v) = W(G)$, i.e. the Wiener index does not change if the vertex $v$ is removed. In 1991, \v{S}olt\'es posed the…

A ring $R$ is called clean if every element of $R$ is the sum of a unit and an idempotent. Motivated by a question proposed by Lam on the cleanness of von Neumann Algebras, Va\v{s} introduced a more natural concept of cleanness for…

A $*$-ring $R$ is called (strongly) $*$-clean if every element of $R$ is the sum of a projection and a unit (which commute with each other). In this note, some properties of $*$-clean rings are considered. In particular, a new class of…