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Let  R  be a finite commutative ring with nonzero identity. The unit graph of  R  is the graph in which the vertex set is  R , and two distinct vertices  x  and  y  are adjacent if and only if  x  +  y  is a unit in  R . In this paper, we determine when these graphs are well-covered, and then, by applying this result, we characterize the unit graphs whose edge rings are Cohen–Macaulay (Gorenstein). This characterization gives us a large class of non-Cohen–Macaulay graphs.

A large class of graphs with a small subclass of Cohen–Macaulay members