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 2n, 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.
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Communicated by László Fuchs
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Akbari, S., Heydari, F. The regular graph of a commutative ring. Period Math Hung 67, 211–220 (2013). https://doi.org/10.1007/s10998-013-7039-1
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DOI: https://doi.org/10.1007/s10998-013-7039-1