The extended zero-divisor graph of a commutative ring. II. (Q1683759)

Let \(R\) be a commutative ring with identity, and let \(Z(R)\) be the set of zero-divisors of \(R.\) The zero-divisor graph of \(R\) is the simple undirected graph with vertex set \(Z^*(R)=Z(R)\setminus \{0\}\) and two distinct vertices \(x\) and \(y\) are adjacent if \(xy=0.\) In this paper, authors define a new graph called extended zero-divisor graph of \(R.\) The extended zero-divisor graph of \(R\) is the undirected simple graph \(\Gamma^\prime(R)\) with vertex set \(Z^*(R)\) and two distinct vertices \(x\) and \(y\) are adjacent if and only if either \(Rx\cap\) Ann\((y)\neq (0)\) or \(Ry\cap\) Ann\((x)\neq (0).\) The new extended zero-divisor graph contains the zero-divisor graph as a spanning subgraph. The zero-divisor graph is connected with diameter at most three. Since the extended zero-divisor graph contains more edges, it is proved that the diameter of the extended zero-divisor graph is at most two and there by the girth is at most four when it contains a cycle. Further authors characterize all commutative rings whose extended zero-divisor graphs are complete or star. It is also proved that when \(R\) is a non-reduced ring, the extended zero-divisor graph is identical with zero-divisor graph.