Geodesic
A graph associated to proper non-small ideals of a commutative ring
Commentationes Mathematicae Universitatis Carolinae, Tome 58 (2017) no. 1, pp. 1-12.

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In this paper, a new kind of graph on a commutative ring is introduced and investigated. Small intersection graph of a ring $R$, denoted by $G(R)$, is a graph with all non-small proper ideals of $R$ as vertices and two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap J$ is not small in $R$. In this article, some interrelation between the graph theoretic properties of this graph and some algebraic properties of rings are studied. We investigated the basic properties of the small intersection graph as diameter, girth, clique number, cut vertex, planar property and independence number. Further, it is shown that the independence number of a small graph of a ring $R$ is equal to the number of its maximal ideals and the domination number of small graph is at most 2.
DOI : 10.14712/1213-7243.2015.189
Classification : 05C25, 05C40, 13A15
Keywords: small ideal; small intersection graph; clique number; independence number; domination number; planar property 
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Atani, S. Ebrahimi; Hesari, S. Dolati Pish; Khoramdel, M. A graph associated to proper non-small ideals of a commutative ring. Commentationes Mathematicae Universitatis Carolinae, Tome 58 (2017) no. 1, pp. 1-12. doi : 10.14712/1213-7243.2015.189. http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.2015.189/

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