Subgraph induced by non-zero idempotent elements of clean graph of some products of fields

Let R be a ring with identity 1 is not equal to 0. The set of all non-zero idempotent elements of a ring R is denoted Id*(R) and the set of all units of a ring R is denoted U(R). The clean graph of R, denoted Cl(R), is a simple graph with vertices of the form (e,u), where e is an idempotent element and u is a unit of R. Moreover, two distinct vertices (e,u) and (f,v) are adjacent if and only if ef = fe = 0 or uv = vu =1. Let Cl_1(R) be the subgraph of Cl(R) induced by {(0,u) | u is element in U(R)} and Cl_2(R) be the subgraph of Cl(R) induced by {(e,u) | e is element of Id*(R) and u is element of U(R)}. In this thesis, we let p and q be prime numbers such that p less than or equal to q and n and m be positive integers. We give two explicit structures of Cl_2(F_p^n x F_q^m). Furthermore, we investigate some basic properties of Cl_2(F_p^n x F_q^m) and Cl(F_p^n x F_q^m) such as minimum degree, maximum degree, and planarity.