On the Genus of the Idempotent Graph of a Finite Commutative Ring

Abstract Let R be a finite commutative ring with identity. The idempotent graph of R is the simple undirected graph I(R) with vertex set, the set of all nontrivial idempotents of R and two distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we have determined all isomorphism classes of finite commutative rings with identity whose I(R) has genus one or two. Also we have determined all isomorphism classes of finite commutative rings with identity whose I(R) has crosscap one. Also we study the the book embedding of toroidal idempotent graphs and classify finite commutative rings whose I(R) is a ring graph.