The Prime Graph PG1(R) of a Ring

Thai Journal of Mathematics, 2019

The notion of a prime graph of a ring $R$, ($PG(R)$) was first introduced by {\sc S. Bhavanari and his coauthors} in [1]. In this paper, we introduce the notion of `Complement of a Prime Graph of a Ring $R$', denote it by $(PG(R))^c$ and find the degree of vertices in $PG(R)$ and $(PG(R))^c$ for the ring $\mz_n$ and the number of triangles in $PG(R)$ and $(PG(R))^c$. It is proved that for any $n \geq 6$ which not a prime then $gr(PG(\mz_n ))=3$. If $n$ is any prime number or $n=4$ then $gr(PG(\mz_n))= \infty$.

In this paper we consider associative rings R (need not be commutative) and defined a new concept 'Prime Graph of R' (denoted by PG(R)). We presented some examples. We obtained few fundamental important results related to PG(R); proved that if R is a semiprime ring, then R is a prime ring if and only if the PG(R) is a tree. Further, we have observed several properties of PG(R) with respect to the properties like: zero divisors, nilpotent elements in R.

Indonesian Journal of Combinatorics

Let R be a finite commutative ring with identity and P be a prime ideal of R. The vertex set is R - {0} and two distinct vertices are adjacent if their product in P. This graph is called the prime ideal graph of R and denoted by ΓP. The relationship among prime ideal, zero-divisor, nilpotent and unit graphs are studied. Also, we show that ΓP is simple connected graph with diameter less than or equal to two and both the clique number and the chromatic number of the graph are equal. Furthermore, it has girth 3 if it contains a cycle. In addition, we compute the number of edges of this graph and investigate some properties of ΓP.

2019

The Prime graph associated to rings denoted by PG(R). PG1(R) is a graph with all the elements of a ring R as vertices, and two distinct vertices x, y are adjacent if and only if either x ·y = 0 or y ·x = 0 or x+y ∈ U(R), the set of all units of R and PG2(R) is a graph with all the elements of a ring R as vertices, and two distinct vertices x, y are adjacent if and only if either x · y = 0 or y · x = 0 or x+ y ∈ Z(R), the set of all zero divisors of R (including zero). In this paper the chromatic number of prime graphs PG1(R) and PG2(R) of ring Zn, where n is power of a prime number, are studied.

2013

This part of the Discussion forms a new bridge between the algebraic concept 'Ring', and 'Graph Theory'. We introduce 'Principal Ideal Graph', denoted by PIG(R), where R is a ring. The concept was introduced by Satyanarayana, Godloza and Nagaraju in the paper 'Some results on Principal ideal Graph of a ring', (published in African Journal of Mathematics and Computer Science Research, vol.4, 2011). We present some examples of PIG(ℤ ℤ ℤ ℤ n ) where ℤ ℤ ℤ ℤ n denotes the ring of integers modulo n for some values of n. We obtain fundamental important relations between rings and graphs with respect to the properties: simple ring, complete graph, Euler graph, etc.

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

BAREKENG: Jurnal Ilmu Matematika dan Terapan

The prime ideal graph of in a finite commutative ring with unity, denoted by , is a graph with elements of as its vertices and two elements in are adjacent if their product is in . In this paper, we explore some interesting properties of . We determined some properties of such as radius, diameter, degree of vertex, girth, clique number, chromatic number, independence number, and domination number. In addition to these properties, we study dimensions of prime ideal graphs, including metric dimension, local metric dimension, and partition dimension; furthermore, we examined topological indices such as atom bond connectivity index, Balaban index, Szeged index, and edge-Szeged index.

JTAM (Jurnal Teori dan Aplikasi Matematika)

Graph theory is a branch of algebra that is growing rapidly both in concept and application studies. This graph application can be used in chemistry, transportation, cryptographic problems, coding theory, design communication network, etc. There is currently a bridge between graphs and algebra, especially an algebraic structures, namely theory of graph algebra. One of researchs on graph algebra is a graph that formed by prime ring elements or called prime graph over ring R. The prime graph over commutative ring R (PG(R))) is a graph construction with set of vertices V(PG(R))=R and two vertices x and y are adjacent if satisfy xRy={0}, for x≠y. Girth is the shortest cycle length contains in PG(R) or can be written gr(PG(R)). Order in PG(R) denoted by |V(PG(R))| and size in PG(R) denoted by |E(PG(R))|. In this paper, we discussed prime graph over cartesian product over rings Z_m×Z_n and its complement. We focused only for m=p_1, n=p_2 and m=p_1, n=〖p_2〗^2, where p_1 and p_2 are prime n...

Acta Universitatis Sapientiae, Mathematica

Let R be a finite commutative ring. We define a co-unit graph, associated to a ring R, denoted by G nu (R) with vertex set V(G nu (R)) = U(R), where U(R) is the set of units of R, and two distinct vertices x, y of U(R) being adjacent if and only if x + y ∉ / U(R). In this paper, we investigate some basic properties of G nu (R), where R is the ring of integers modulo n, for different values of n. We find the domination number, clique number and the girth of G nu (R).

2020

Let R be a ring, we associate a simple graph Φ(R) to R, with vertices V (R) = R\{0, 1,−1}, where distinct vertices x, y ∈ V (R) are adjacent if and only if either xy ̸= 0 or yx ̸= 0. In this paper, we prove that if Φ(R) is connected such that R Z2×Z4 then the diameter of Φ(R) is almost 2. Also, we will pay specific attention to investigate the connectivity of certain rings such that, the ring of integers modulo n,Zn is connected, reduced ring and matrix ring.