Well-covered unitary Cayley graphs of matrix rings over finite fields and applications

Abstract

Suppose that F is a finite field and R=Mn(F) is the ring of n-square matrices over F. Here we characterize when the Cayley graph of the additive group of R with respect to the set of invertible elements of R, called the unitary Cayley graph of R, is well-covered. Then we apply this to characterize all finite rings with identity whose unitary Cayley graph is well-covered or Cohen-Macaulay.