THE EMBEDDING OF LINE GRAPHS ASSOCIATED TO JACOBSON GRAPHS IN SPHERE AND PROJECTIVE PLANE

Let R be a commutative ring with non-zero identity and J(R) be the Jacobson radical of R. The Jacobson graph of R, denoted by JR, is a graph with vertex set R n J(R),such that two distinct vertices a and b in R n J(R) are adjacent if and only if 1 - ab is not a unit of R. Also, the line graph of the Jacobson graph is denoted by L (JR). In this paper, we characterize all finite commutative rings R with non-zero identity (up to isomorphism) such that the graphs L (JR) are planar or projective.