Classification of Finite Commutative Rings with Planar, Toroidal, and Projective Line Graphs Associated with Jacobson Graphs

Let R be a commutative ring with nonzero identity and J(R) be the Jacobson radical of R. The Jacobson graph of R, denoted by J(R), is a graph with vertex-set R \ J(R), such that two distinct vertices a and b in R \ 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(J(R)). In this paper, we characterize all finite commutative rings R such that the graphs L(J(R)) are planar, toroidal or projective.