Structural Properties of the Essential Ideal Graph of <img src=image/13424240_01.gif>

Abstract

Let S be a commutative ring with unity. The essential ideal graph of S, denoted by E S , is a graph with vertex set consisting of all nonzero proper ideals of A and two vertices P and Q are adjacent whenever P + Q is an essential ideal. An essential ideal P of a ring S is an ideal P of S (P S), having nonzero intersection with every other ideal of S. The set M ax(S) contains all the maximal ideals of S. The Jacobson radical of S, J(S), is the set of intersection of all maximal ideals of S. The comaximal ideal graph of S, denoted by C(S), is a simple graph with vertices as proper ideals of A not contained in J(S) and the vertices P and Q are associated with an edge whenever P + Q = S. In this paper, we study the structural properties of the graph E S by using the ring theoretic concepts. We obtain a characterization for E S to be isomorphic to the comaximal ideal graph C(S). Moreover, we derive the structure theorem of E Zn and determine graph parameters like clique number, chromatic number and independence number. Also, we characterize the perfectness of E Zn and determine the values of n for which E Zn 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 E Zn for some n.

Adjacency Spectrum and Wiener Index of the Essential Ideal Graph of a Finite Commutative Ring $\mathbb{Z}_{n}$

“…Let n = 36 = 2 2 3 2 . The vertices of E Z 36 are (2), ( 3), ( 4), ( 6), ( 9), (12), and (18) and can be partitioned as follows: 9), (18)}. Since X contains all the proper essential ideals of Z 36 , the subraph…”