On the essential annihilating-ideal graph of commutative rings - Nazim, Mohd (Ars mathematica contemporanea, 2022)
Let ?$R$? be a commutative ring with unity, ?$A(R)$? be the set of all annihilating-ideals of ?$R$? and ?$A^{*}(R) = A(R) \setminus \{0\}$?. In this paper, we introduced and studied the essential annihilating-ideal graph of ?$R$?, denoted by ?$\mathcal{EG}(R)$?, with vertex set ?$A^{*}(R)$? and two distinct vertices ?$I_1$? and ?$I_2$? are adjacent if and only if v ?$Ann (I_1 I_2)$? is an essential ideal of ?$R$?. We prove that ?$\mathcal{EG}(R)$? is a connected graph with diameter at most three and girth at most four if ?$\mathcal{EG}(R)$? contains a cycle. Furthermore, the rings ?$R$? are characterized for which ?$\mathcal{EG}(R)$? is a star or a complete graph. Finally, we classify all the Artinian rings ?$R$? for which ?$\mathcal{EG}(R)$? is isomorphic to some well-known graphs.
Keywords: annihilating-ideal graph; complete graph; genus of a graph; graf anihilirajočih idealov; graf deliteljev niča; planar graph; polni graf; ravninski graf; rod grafa; zero-divisor graph