On the planarity, genus and crosscap of new extension of zero-divisor graph of commutative rings

Let R be a commutative ring with Z(R) its set of all zero-divisors. The new extension of zero-divisor graph of a commutative ring R denoted by Γ˜(R), is an undirected graph with vertex set Z(R)* and two vertices x and y are adjacent if and only if either xy = 0 or x+y∈Z(R). In this paper, first we characterized finite rings R for which Γ˜(R), is isomorphic to some well-known graphs and then we classify all the finite rings R for which Γ˜(R) is planar, outerplanar, toroidal or double toroidal. Finally, we classify the finite rings R for which the graph Γ˜(R) has crosscap at most two.