An extension of the essential graph of a ring

Abstract

Let A be a commutative ring with non-zero identity, and E(A)={p∈A|annA(pq)≤eA,for some q∈A∗}.The extended essential graph, denoted by EgG(A) is a graph with the vertex set E(A)∗=E(A)\{0}. Two distinct vertices r,s∈E(A)∗ are adjacent if and only if annA(rs)≤eA. In this paper, we prove that EgG(A) is connected with diam(EgG(A))≤3 and if EgG(A) has a cycle, the ngr(EgG(A))≤4. Furthermore, we establishthat if A is an Artinian commutative ring, then ω(EgG(A))=χ(EgG(A))=|N(A)∗|+|Max(A)|.