Let P be a finite commutative ring with identity, and let Z (P) denote the set of its zero-divisors. The extended zero-divisor graph of P, denoted by Γ˜ (P), is the undirected simple graph with vertex set Z (P) *=Z (P) ∖0, where two distinct vertices α and β are adjacent if and only if αβ=0 or α+β∈Z (P). For a graph G, let L (G) denote its line graph. In this paper, we first characterize all finite commutative rings P for which Γ˜ (P) is a line graph of some graph. We then classify the finite commutative rings P such that L (Γ˜ (P) ) is planar, outerplanar, or 2-outerplanar. Finally, we obtain a complete classification of finite commutative rings P for which L (Γ˜ (P) ) is toroidal.
Raza et al. (Mon,) studied this question.