# The regular graph of a commutative ring

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## Abstract

Let *R* be a commutative ring, let *Z*(*R*) be the set of all zero-divisors of *R* and Reg(*R*) = *R*\*Z*(*R*). The *regular graph* of *R*, denoted by *G*(*R*), is a graph with all elements of Reg(*R*) as the vertices, and two distinct vertices *x, y* ∈ Reg(*R*) are adjacent if and only if *x+y* ∈ *Z*(*R*). In this paper we show that 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 2^{n}, where *n* is the minimum number of prime ideals whose union is *Z*(*R*). Also, we prove that all trees that can occur as the regular graph of a ring have at most two vertices.

### Key words and phrases

regular graph Noetherian ring zero-divisors clique number chromatic number### Mathematics subject classification numbers

05C15 05C25 05C69 13A 13E05## Preview

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### References

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© Akadémiai Kiadó, Budapest, Hungary 2013