# The total graph and regular graph of a commutative ring

@article{Akbari2009TheTG, title={The total graph and regular graph of a commutative ring}, author={Saieed Akbari and Dariush Kiani and Fatemeh Mohammadi and Somayeh Moradi}, journal={Journal of Pure and Applied Algebra}, year={2009}, volume={213}, pages={2224-2228} }

## 111 Citations

### 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,…

### 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 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…

### The M-Regular Graph of a Commutative Ring

- Mathematics
- 2013

Abstract Let R be a commutative ring and M be an R-module, and let Z(M) be the set of all zero-divisors on M. In 2008, D. F. Anderson and A. Badawi introduced the regular graph of R. In this paper,…

### MINIMUM FLOWS IN THE TOTAL GRAPH OF A FINITE COMMUTATIVE RING

- Mathematics
- 2014

Let R be a commutative ring with zero-divisor set Z(R). The total graph of R, denoted by T (Γ(R)), is the simple (undirected) graph with vertex set R where two distinct vertices are adjacent if their…

### Some Properties of a Cayley Graph of a Commutative Ring

- Mathematics
- 2014

Let R be a commutative ring with unity and R +, U(R), and Z*(R) be the additive group, the set of unit elements, and the set of all nonzero zero-divisors of R, respectively. We denote by ℂ𝔸𝕐(R) and…

### SOME RESULTS ON COZERO-DIVISOR GRAPH OF A COMMUTATIVE RING

- Mathematics
- 2014

Let R be a commutative ring with unity. The cozero-divisor graph of R denoted by Γ′(R) is a graph with the vertex set W*(R), where W*(R) is the set of all non-zero and non-unit elements of R, and two…

### On the Cayley Graph of a Commutative Ring with Respect to its Zero-divisors

- Mathematics
- 2013

Let R be a commutative ring with unity and R+ and Z*(R) be the additive group and the set of all nonzero zero-divisors of R, respectively. We denote by ℂ𝔸𝕐(R) the Cayley graph Cay(R+, Z*(R)). In…

## References

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- 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).

### Introduction to commutative algebra

- Mathematics
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* Introduction * Rings and Ideals * Modules * Rings and Modules of Fractions * Primary Decomposition * Integral Dependence and Valuations * Chain Conditions * Noetherian Rings * Artin Rings *…