The embedding of line graphs associated to the zero-divisor graphs of commutative rings (Q616521)

Let \(R\) be a commutative ring with nonzero identity, and let \(\text{Z}(R)\) be its set of zero divisors. The zero-divisor graph, \(\Gamma(R)\), is the graph with vertices the set of nonzero zero divisors of \(R\), and for distinct \(x,y \in {Z(R)}\), the vertices \(x\) and \(y\) are adjacent if and only if \(xy=0\). In the paper under review, the authors study the minimal embedding of the line graph associated to \(\Gamma(R)\) into compact surfaces (orientable or non-orientable) and completely classify all finite commutative rings \(R\) such that the line graphs associated to their zero-divisor graphs have genera or crosscaps up to two.