On the line graph structure of the cozero-divisor graph of a commutative ring

Mojgan Afkhami () and Zahra Barati ()
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Mojgan Afkhami: University of Neyshabur
Zahra Barati: Kosar University of Bojnord

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 3, 1193-1199

Abstract: Abstract Assume that R is a commutative ring with non-zero identity and $$W^*(R)$$ W ∗ ( R ) is the set of all non-zero non-unit elements of R. Also, for $$x\in R$$ x ∈ R , the ideal which is generated by x, is denoted by Rx. The cozero-divisor graph of R, which is denoted by $$\Gamma '(R)$$ Γ ′ ( R ) , is a graph with $$W^*(R)$$ W ∗ ( R ) as the vertex-set, and two distinct vertices x and y are adjacent in $$W^*(R)$$ W ∗ ( R ) if and only if $$x\notin Ry$$ x ∉ R y and $$y\notin Rx$$ y ∉ R x . In this paper, we completely determine all finite commutative rings R such that $$\Gamma '(R)$$ Γ ′ ( R ) is a line graph. We also characterize all finite commutative rings R such that $$\Gamma '(R)$$ Γ ′ ( R ) is isomorphic to its line graph.

Keywords: Cozero-divisor graph; Line graph; Commutative ring; 05C25; 05C75; 13A99 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-024-00568-6

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