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 $\mathfrak{J}_R$, is a graph with vertex-set $ R \setminus J(R)$, such that two distinct vertices $a$ and $b$ in $R\setminus J(R)$ 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 $L(\mathfrak{J}_R)$. In this paper, we characterize all finite commutative rings $R$ such that the graphs $L(\mathfrak{J}_R)$ are planar, toroidal or projective.