Well-covered Unit Graphs of Finite Rings

Let R be a finite ring with identity. The unit graph (unitary Cayley graph) of R is the graph with vertex set R, where two distinct vertices x and y are adjacent exactly whenever x+y is a unit in R (x-y is a unit in R). Here, we study independent sets of unit graphs of matrix rings over finite fields and use them to characterize all finite rings for which the unit graph is well-covered or Cohen-Macaulay. Moreover, we show that the unit graph of R is well-covered if and only if the unitary Cayley graph of R is well-covered and the characteristic of R/J (R)