Strong Resolving Graphs of U ‐Clean Graphs of Finite Commutative Rings

Let R be a finite commutative ring with identity 1. The U ‐clean graph U ‐ C l ( R ) of a ring R is a simple undirected graph with vertices are of the form ( e , u ), where e is a nonzero idempotent and u is a unit of R , and two distinct vertices ( e , u ), ( f , v ) of U ‐ C l ( R ) are adjacent if and only if e = f = 1 or u v = 1. In this paper, we present the strong resolving graph U ‐ C l ( R ) S R of U ‐ C l ( R ). The vertex degrees, the independence number, the clique number, and the chromatic number of U ‐ C l ( R ) S R are determined. Moreover, we show that if k is a divisor of ( n − 1) n , where k and n are positive integers with k > n ≥ 3, then U ‐ is not a complete graph. Finally, we give some examples of U ‐ and U ‐ to illustrate our results.