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

Abstract

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 is an undirected graph with vertex set and two vertices x and y are adjacent if and only if either xy = 0 or In this paper, first we characterized finite rings R for which is isomorphic to some well-known graphs and then we classify all the finite rings R for which is planar, outerplanar, toroidal or double toroidal. Finally, we classify the finite rings R for which the graph has crosscap at most two.