S. Ebrahimi Atani, S. Dolati Pish Hesari, M. Khoramdel
A graph associated to proper non-small ideals of a commutative ring

Comment.Math.Univ.Carolin. 58,1 (2017) 1-12.

Abstract: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.

Keywords: small ideal; small intersection graph; clique number; independence number; domination number; planar property

DOI: DOI 10.14712/1213-7243.2015.189
AMS Subject Classification: 05C40 05C25 13A15