On the Clean Graph of Commutative Artinian Rings

For a commutative Artinian ring R with unity, the clean graph C l ( R ) is a graph with vertices in the form of an ordered pair ( e , u ), where e is an idempotent and u is a unit of ring R , respectively. Two distinct vertices ( e , u ) and ( f , v ) are adjacent in C l ( R ) if and only if e f = f e = 0 or u v = v u = 1. In this study, we consider C l 2 ( R ) as the subgraph of C l ( R ) induced by e is a non − zero idempotent element of R }. We show that C l 2 ( R ) contains the Hamiltonian cycle. Also, we compute the graphic sequence, matching number, vertex cover number, edge cover number, vertex connectivity, and edge connectivity of C l 2 ( R ). As an application, we compute the Wiener and the first Zagreb indices of C l 2 ( R ). Moreover, we give the formula for the number of self invertible elements of .