Simple-intersection graphs of rings

Fida Moh'd; Mamoon Ahmed; Fida Moh'd; Mamoon Ahmed

doi:10.3934/math.2023051

AIMS Mathematics

2023, Volume 8, Issue 1: 1040-1054. doi: 10.3934/math.2023051

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Research article

Simple-intersection graphs of rings

Fida Moh'd ,
Mamoon Ahmed ^,

Department of Basic Sciences, Princess Sumaya University for Technology, Amman, Jordan

Received: 08 July 2022 Revised: 07 October 2022 Accepted: 08 October 2022 Published: 14 October 2022
MSC : 05C07, 05C25, 05C38, 05C40, 05C69

Let $ R $ be a ring with unity. In this paper, we introduce a new graph associated with $ R $ called the simple-intersection graph of $ R $, denoted by $ GS(R) $. The vertices of $ GS(R) $ are the nonzero ideals of $ R $, and two vertices are adjacent if and only if their intersection is a nonzero simple ideal. We study the interplay between the algebraic properties of $ R $, and the graph properties of $ GS(R) $ such as connectedness, bipartiteness, girth, dominating sets, etc. Moreover, we determine the precise values of the girth and diameter of $ GS(R) $, as well as give a formula to compute the clique and domination numbers of $ GS(R) $.
- simple-intersection graph,
- rings,
- ideals,
- clique,
- girth,
- dominating number,
- cycles,
- diameter,
- connected graph,
- complete graph
Citation: Fida Moh'd, Mamoon Ahmed. Simple-intersection graphs of rings[J]. AIMS Mathematics, 2023, 8(1): 1040-1054. doi: 10.3934/math.2023051

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Abstract

Let $ R $ be a ring with unity. In this paper, we introduce a new graph associated with $ R $ called the simple-intersection graph of $ R $, denoted by $ GS(R) $. The vertices of $ GS(R) $ are the nonzero ideals of $ R $, and two vertices are adjacent if and only if their intersection is a nonzero simple ideal. We study the interplay between the algebraic properties of $ R $, and the graph properties of $ GS(R) $ such as connectedness, bipartiteness, girth, dominating sets, etc. Moreover, we determine the precise values of the girth and diameter of $ GS(R) $, as well as give a formula to compute the clique and domination numbers of $ GS(R) $.

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