Classification of finite commutative rings with planar, toroidal, and projective line graphs associated with Jacobson graphs (Q268085)

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 \(R\backslash J(R)\), such that two distinct vertices \(a\) and \(b\) in \(R\backslash J(R)\) 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 \(L(\mathfrak{J}_R)\). The main results in the paper under review indicates that if \(R\) is a finite nonlocal ring, then (1) the graph \(L(\mathfrak{J}_R)\) is planar if and only if \(R\) is isomorphic to \(\mathbb{Z}_2\times\mathbb{Z}_2\) or \(\mathbb{Z}_2\times\mathbb{Z}_3\); (2) the graph \(L(\mathfrak{J}_R)\) is toroidal if and only if \(R\) is isomorphic to \(\mathbb{Z}_2\times\mathbb{F}_4\) or \(\mathbb{Z}_3\times\mathbb{Z}_3\); (3) the graph \(L(\mathfrak{J}_R)\) is projective if and only if \(R\) is isomorphic to \(\mathbb{Z}_2\times\mathbb{F}_4\), where \(\mathbb{F}_4\) is a field with 4 elements. Also if \((R,\mathfrak{m})\) is a finite local ring, then the graph \(L(\mathfrak{J}_R)\) is neither toroidal nor projective and is planar if and only if \(R\) isomorphic to one of \(\mathbb{Z}_2[x]/(x^2)\), \(\mathbb{Z}_4\), \(\mathbb{Z}_2[x]/(x^3)\), \(\mathbb{Z}_4[x]/(2x,x^2)\), \(\mathbb{Z}_4[x]/(x^2-2,x^3)\), \(\mathbb{Z}_8\), \(\mathbb{Z}_3[x]/(x^2)\), or \(\mathbb{Z}_9\).