Korean J. Math. Vol. 30 No. 1 (2022) pp.53-60
DOI: https://doi.org/10.11568/kjm.2022.30.1.53

Line graphs of unit graphs associated with the direct product of rings

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Published: Mar 30, 2022
Keywords:
Ring of integers modulo, zero-divisor graph, unit graph, line graph
Mathematics Subject Classification:
13A99, 05C12 05C25, 05C69, 05C78

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Shariefuddin Pirzada
Department of Mathematics, University of Kashmir, Srinagar, 190006, Srinagar, Kashmir, India.
Aaqib Altaf
Department of Mathematics, University of Kashmir, Srinagar, 190006, Kashmir, India.

Abstract

Let $R$ be a finite commutative ring with non zero identity. The unit graph of $R$ denoted by $G(R)$ is the graph obtained by setting all the elements of $R$ to be the vertices of a graph and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y \in U(R)$, where $U(R)$ denotes the set of units of $R$. In this paper, we find the commutative rings $R$ for which $G(R)$ is a line graph. Also, we find the rings for which the complements of unit graphs are line graphs.


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