## 284 Citations

### ON THE TOTAL GRAPH OF A COMMUTATIVE RING WITHOUT THE ZERO ELEMENT

- Mathematics
- 2012

Let R be a commutative ring with nonzero identity, and let Z(R) be its set of zero-divisors. The total graph of R is the (undirected) graph T(Γ(R)) with vertices all elements of R, and two distinct…

### The total graph of a finite commutative ring

- Mathematics
- 2013

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,…

### The regular graph of a commutative ring

- MathematicsPeriod. Math. Hung.
- 2013

If R is a commutative Noetherian ring and 2 ∈ Z(R), then the chromatic number and the clique number of G(R) are the same and they are 2n, where n is the minimum number of prime ideals whose union is Z( R).

### When a total graph associated with a commutative ring is perfect?

- MathematicsPublications de l'Institut Math?matique (Belgrade)
- 2020

Let R be a commutative ring with identity, and let Z(R) be the set of
zero-divisors of R. The total graph of R is the graph T(?(R)) whose vertices
are all elements of R, and two distinct vertices x…

### Total Zero Divisor Graph of a Commutative Ring

- Mathematics
- 2012

ABSTRACT Let R be a commutative ring with Z(R), its set of zero divisors. The total zero divisor graph of R, denoted Z(Γ(R)) is the undirected (simple) graph with vertices Z(R)*=Z(R)-{0}, the set of…

### The Girth of the Total Graph of ℤ n

- MathematicsProceedings of the 1st International Conference on Mathematics and Mathematics Education (ICMMEd 2020)
- 2021

Let R be a commutative ring with a non-zero identity, and Z(R) is a set of zero-divisors of R. The total graph of R, denoted TΓ(R), is an (undirected) graph with all elements R as vertices of TΓ(R)…

### On the Total Graph and Its Complement of a Commutative Ring

- Mathematics
- 2013

Let R be a commutative ring and Z(R) be its set of all zero-divisors. The total graph of R, denoted by T Γ(R), is the undirected graph with vertex set R and two distinct vertices x and y are adjacent…

### On the nilpotent graph of a ring

- MathematicsTurkish Journal of Mathematics
- 2013

Let R be a ring with unity. The nilpotent graph of R , denoted by ΓN (R) , is a graph with vertex set ZN (R) ∗ = {0 � x ∈ R | xy ∈ N (R) for some 0 � y ∈ R} ; and two distinct vertices x and y are…

## References

SHOWING 1-10 OF 15 REFERENCES

### On the Zero-Divisor Graph of a Ring

- Mathematics
- 2008

Let R be a commutative ring with identity, Z(R) its set of zero-divisors, and Nil(R) its ideal of nilpotent elements. The zero-divisor graph of R is Γ(R) = Z(R)\{0}, with distinct vertices x and y…

### Infinite Planar Zero-Divisor Graphs

- Mathematics
- 2006

Given a commutative ring R, one can associate with R an undirected graph Γ(R) whose vertices are the nonzero zero-divisors of R, and two distinct vertices x and y are joined by an edge iff xy = 0. In…

### The Zero-Divisor Graph of a Commutative Ring☆

- Mathematics
- 1999

For each commutative ring R we associate a (simple) graph Γ(R). We investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of Γ(R).

### Zero-Divisor Graphs of Polynomials and Power Series Over Commutative Rings

- Mathematics
- 2005

ABSTRACT We recall several results about zero-divisor graphs of commutative rings. Then we examine the preservation of diameter and girth of the zero-divisor graph under extension to polynomial and…