An extension of the essential graph of a ring

Abstract

Let \(A\) be a commutative ring with non-zero identity, and \(E(A)=\{p\in A | ann_A(pq)\leq_e A, ~\mbox { for some } ~q\in A^* \}\) . The extended essential graph, denoted by \(E_gG(A)\) is a graph with the vertex set \(E(A)^*=E(A)\setminus\{0\}\). Two distinct vertices \(r, s\in E(A)^*\) are adjacent if and only if \(ann_A(rs)\leq_e A\). In this paper, we prove that \(E_gG(A)\) is connected with \(diam(E_gG(A))\leq 3\) and if \(E_gG(A)\) has a cycle, then \(gr(E_gG(A))\leq 4\). Furthermore, we establish that if \(A\) is an Artinian commutative ring, then \(\omega (E_gG(A))=\chi (E_gG(A))=|N(A)^*|+ |Max(A)|\).