### Journals Information

**
Mathematics and Statistics Vol. 9(5), pp. 806 - 815 DOI: 10.13189/ms.2021.090521 Reprint (PDF) (384Kb) **

## Structural Properties of the Essential Ideal Graph of

**P Jamsheena ^{*}, A V Chithra **

Department of Mathematics, National Institute of Technology Calicut, Kozhikode, 673601, Kerala, India

**ABSTRACT**

Let be a commutative ring with unity. The essential ideal graph of , denoted by , is a graph with vertex set consisting of all nonzero proper ideals of A and two vertices and are adjacent whenever is an essential ideal. An essential ideal of a ring is an ideal of (), having nonzero intersection with every other ideal of . The set contains all the maximal ideals of . The Jacobson radical of , , is the set of intersection of all maximal ideals of . The comaximal ideal graph of , denoted by , is a simple graph with vertices as proper ideals of A not contained in and the vertices and are associated with an edge whenever . In this paper, we study the structural properties of the graph by using the ring theoretic concepts. We obtain a characterization for to be isomorphic to the comaximal ideal graph . Moreover, we derive the structure theorem of and determine graph parameters like clique number, chromatic number and independence number. Also, we characterize the perfectness of and determine the values of for which is split and claw-free, Eulerian and Hamiltonian. In addition, we show that the finite essential ideal graph of any non-local ring is isomorphic to for some .

**KEYWORDS**

Essential Ideal Graph of a Commutative Ring, Co-maximal Ideal Graph, Matching, Perfect Graph, Clique Number, Chromatic Number

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] P Jamsheena , A V Chithra , "Structural Properties of the Essential Ideal Graph of ," Mathematics and Statistics, Vol. 9, No. 5, pp. 806 - 815, 2021. DOI: 10.13189/ms.2021.090521.

(b). APA Format:

P Jamsheena , A V Chithra (2021). Structural Properties of the Essential Ideal Graph of . Mathematics and Statistics, 9(5), 806 - 815. DOI: 10.13189/ms.2021.090521.