Line Graphs and Embedding Properties Associated with Extended Zero-Divisor Graph of Commutative Rings

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.