Classification of Rings Associated with the Genus of Clean Graphs

Let [Formula: see text] be a ring (not necessarily commutative) with identity 1 and let [Formula: see text] be its clean graph. In this paper, we investigate the genus number of the compact Riemann surface in which [Formula: see text] can be embedded and explicitly determine all commutative rings [Formula: see text] (up to isomorphism) such that [Formula: see text] has genus at most two. It is shown that for any Artinian ring [Formula: see text], [Formula: see text] is a projective graph if and only if [Formula: see text] is isomorphic to [Formula: see text]. Furthermore, we determine all isomorphism classes of commutative rings whose clean graphs have crosscap two.