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Publication

The n-zero-divisor graph of a commutative semigroup

Anderson, David F.
Badawi, Ayman
Date
2022-04-16
Advisor
Type
Article
Peer-Reviewed
Postprint
Degree
Description
Abstract
Let S be a (multiplicative) commutative semigroup with 0, Z(S) the set of zero-divisors of S, and n a positive integer. The zero-divisor graph of S is the (simple) graph Γ(S) with vertices Z(S) ∗ = Z(S) \ {0}, and distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we introduce and study the n-zero-divisor graph of S as the (simple) graph Γn(S) with vertices Zn(S) ∗ = {x n | x ∈ Z(S)} \ {0}, and distinct vertices x and y are adjacent if and only if xy = 0. Thus each Γn(S) is an induced subgraph of Γ(S) = Γ1(S). We pay particular attention to diam(Γn(S)), gr(Γn(S)), and the case when S is a commutative ring with 1 6= 0. We also consider several other types of “n-zero-divisor” graphs and commutative rings such that some power of every element (or zero-divisor) is idempotent.