**Ideal Graphs Supported
By Given Ideals of Commutative Rings**

F. H. Abdulqadr^{1}

^{1}Mathematics Department, College of Education, Salahaddin University, Erbil-Iraq**Original:** 9 November 2019 ** ****Revised**: 12 January 2019 ** ****Accepted**: 30 January 2020 **Published online**: 20 June 2020

**Doi Link**; https://doi.org/10.17656/jzs.10786

**Abstract**

In this paper we introduce and study a new kind of graph that constructed by non-trivial ideals of a commutative ring with identity. Let R be a commutative ring with identity and P be a non-trivial ideal of R. The ideal graph supported by the ideal P, denoted by (P), is the undirected graph whose vertices are those non-trivial ideals I of R such that there exists a non-trivial ideal JI of R with IJ⊂P, and every two vertices I and J are adjacent if IJ and IJ⊂P. We investigate the connectivity, completeness and planarity of the graph (P). Also we explore the diameter, girth, domination, clique number and chromatic number of (P).

**Key Words:**

*Ideal graph supported by given ideals of commutative rings, connected graphs , Clique and chromatic number.*

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