The total graph of a finite commutative ring
نویسنده
چکیده
Let R be a commutative ring with Z(R) , its set of zero-divisors and Reg(R) , its set of regular elements. Total graph of R , denoted by T (Γ(R)) , is the graph with all elements of R as vertices, and two distinct vertices x, y ∈ R , are adjacent in T (Γ(R)) if and only if x+ y ∈ Z(R) . In this paper, some properties of T (Γ(R)) have been investigated, where R is a finite commutative ring and a new upper bound for vertex-connectivity has been obtained in this case. Also, we have proved that the edge-connectivity of T (Γ(R)) coincides with the minimum degree if and only if R is a finite commutative ring such that Z(R) is not an ideal in R .
منابع مشابه
On the Zero-divisor Cayley Graph of a Finite Commutative Ring
Let R be a fnite commutative ring and N(R) be the set of non unit elements of R. The non unit graph of R, denoted by Gamma(R), is the graph obtained by setting all the elements of N(R) to be the vertices and defning distinct vertices x and y to be adjacent if and only if x - yin N(R). In this paper, the basic properties of Gamma(R) are investigated and some characterization results regarding co...
متن کاملON THE REFINEMENT OF THE UNIT AND UNITARY CAYLEY GRAPHS OF RINGS
Let $R$ be a ring (not necessarily commutative) with nonzero identity. We define $Gamma(R)$ to be the graph with vertex set $R$ in which two distinct vertices $x$ and $y$ are adjacent if and only if there exist unit elements $u,v$ of $R$ such that $x+uyv$ is a unit of $R$. In this paper, basic properties of $Gamma(R)$ are studied. We investigate connectivity and the girth of $Gamma(R)$, where $...
متن کاملNILPOTENT GRAPHS OF MATRIX ALGEBRAS
Let $R$ be a ring with unity. The undirected nilpotent graph of $R$, denoted by $Gamma_N(R)$, is a graph with vertex set ~$Z_N(R)^* = {0neq x in R | xy in N(R) for some y in R^*}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy in N(R)$, or equivalently, $yx in N(R)$, where $N(R)$ denoted the nilpotent elements of $R$. Recently, it has been proved that if $R$ is a left A...
متن کاملA note on a graph related to the comaximal ideal graph of a commutative ring
The rings considered in this article are commutative with identity which admit at least two maximal ideals. This article is inspired by the work done on the comaximal ideal graph of a commutative ring. Let R be a ring. We associate an undirected graph to R denoted by mathcal{G}(R), whose vertex set is the set of all proper ideals I of R such that Inotsubseteq J(R), where J(R) is...
متن کاملStudy on the new graph constructed by a commutative ring
Let R be a commutative ring and G(R) be a graph with vertices as proper andnon-trivial ideals of R. Two distinct vertices I and J are said to be adjacentif and only if I + J = R. In this paper we study a graph constructed froma subgraph G(R)Δ(R) of G(R) which consists of all ideals I of R such thatI Δ J(R), where J(R) denotes the Jacobson radical of R. In this paper westudy about the relation b...
متن کاملA graph associated to spectrum of a commutative ring
Let $R$ be a commutative ring. In this paper, by using algebraic properties of $R$, we study the Hase digraph of prime ideals of $R$.
متن کامل