Exploring the Embedding of the Extended Zero-Divisor Graph of Commutative Rings

Rc represents commutative rings that have unity elements. The collection of all zero-divisor elements in Rc are represented by D(Rc). We denote an extended zero-divisor graph by the notation ℸ′(Rc) of Rc. This graph has a set of vertices in D(Rc)*. The graph ℸ′(Rc) illustrates interactions among the zero-divisor elements of Rc. Specifically, two different vertices u and y are connected in ℸ′(Rc) iff uRc∩Ann(y) is non-null or yRc∩Ann(u) is non-null. The main idea for this work is to systematically analyze the ring Rc which is finite for the unique aspect of their extended zero-divisor graph. This study particularly focuses on instances where the extended zero-divisor graph has a genus or crosscap of two. Furthermore, this work aims to thoroughly characterize finite ring Rc wherein the extended zero-divisor graph ℸ′(Rc) has an outerplanarity index of two. Finally, we determine the book thickness of ℸ′(Rc) for genus at most one.