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-ClR 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-ClR are adjacent if and only if e=f=1 or uv=1. In this paper, we present the strong resolving graph U-ClRSR of U-ClR. The vertex degrees, the independence number, the clique number, and the chromatic number of U-ClRSR are determined. Moreover, we show that if k is a divisor of n−1n, where k and n are positive integers with k>n≥3, then U-ClZk is not a complete graph. Finally, we give some examples of U-ClZn and U-ClSRZn to illustrate our results.