Zero-Divisor Graph in Power Set Ring and Matrices Ring of 2 X 2

The first article about graph theory was written by Leonhard Euler the famous Swiss mathematician which was published in 1736. Primarily, the idea of graph was not important as point of mathematics because it most deals with recreational puzzles. But the recent improvement in mathematics specially its application brought a strong revolution in graph theory. Therefore, this article is written under the tile of (Zero-divisor graph in power set ring and matrices ring of). In this article we first introduce the zero-divisor graph set of alternative in ring. Then we present the zero-divisor graph in power set ring and matrices ring of. Whereas the vertex is denoted with and the elements of an optional ring which are not zero-divisors they are the vertices without edges of that ring in zero-divisor graph. Next, we will study when a graph is planar in power set ring and under which circumstances a graph is complete and complete bipartite in matrices set ring of. After the research we found out: If the element numbers of set be minor than 4, the zero-divisor graph ring of is planar and if the numbers of X set elements be maximum than 4, the zero-divisor graph ring is not planar. Suppose that matrix ring of is in field with elements. If, The sub inductive graph on is complete bipartite of zero-divisor graph in ring and if or, in this case the zero-divisor graph in ring of sub inductive graph on is complete. The zero-divisor graphs in ring are planar if and only if.