Isomorphism classes of clean graphs of rings of integers modulo N

Let R be a ring with identity 1 , 0. The set of all idempotent elementsof R is denoted by Id(R) and the set of all units of R is denoted by U(R). The cleangraph of R, denoted by Cl(R), is a simple graph with vertices of the form (e,u), wheree ∈ Id(R) and u ∈ U(R). Two distinct vertices (e1,u1) and (e2,u2) are adjacent if andonly if e1e2 = e2e1 = 0 or u1u2 = u2u1 = 1. In this thesis, we first investigate Cl2(R),the subgraph of Cl(R) induced by the set {(e,u) | e ∈ Id(R)\{0} and u ∈ U(R)}.For any distinct prime numbers p,q and positive integers n,m,r such that gcd(p,r) =gcd(q,r) = 1, we give the necessary and sufficient conditions for isomorphism betweenCl2(Zpnr) and Cl2(Zqmr). Moreover, we present the necessary and sufficient conditionsfor isomorphism between Cl(Zpnr) and Cl(Zqmr).