The zero-divisor graph of a commutative ring $R$ with respect to nilpotent elements is a simple undirected graph $\Gamma_N^*(R)$ with vertex set $\mathcal{Z}_N(R)^*$, and two vertices $x$ and $y$ are adjacent if and only if $xy$ is nilpotent and $xy\neq 0$, where $\mathcal{Z}_N(R)=\{x\in R: xy~\text{is nilpotent, for some} y\in R^*\}$. In this paper, we investigate the basic properties of $\Gamma_N^*(R)$. We discuss when it will be Eulerian and Hamiltonian. We further determine the genus of $\Gamma_N^*(R)$.

D. F. Anderson, A. Frazier, A. Lauve and P. S. Livingston (2001). The zero-divisor graph of a commutative ring II. Lecture Notes in Pure and Appl. Math.. 220, 61-72

2

D. F. Anderson and P. S. Livingston (1999). The zero-divisor graph of a commutative ring. J. Algebra. 217, 434-447

3

M. F. Atiyah and I. G. Macdonald (1969). Introduction to Commutative Algebra. Addison-Wesley Publishing Company, Reading, Mass.-London-Don Mills.

4

I. Beck (1988). Coloring of commutative rings. J. Algebra. 116, 208-226

5

R. Belshoff and J. Chapman (2007). Planar zero-divisor graphs. J. Algebra. 316, 471-480

6

G. Chartrand and L. Lesniak (1986). Graphs and Digraphs. Wadsworth and Brooks/ Cole, Monterey, CA.

7

P. W. Chen (2003). A kind of graph structure of rings. Algebra Colloq.. 10, 229-238

8

S. F"oldes and P. L. Hammer (1977). Split graphs. Congressus Numerantium. 19, 311-315

9

R. Kala and S. Kavitha (2014). Nilpotent graph of genus one. Discrete Math. Algorithms Appl.. 6 (3), 10

10

A. Li and Q. Li (2010). A kind of graph of structure on von-Neumann regular rings. Inter. J. Algebra. 4 (6), 291-302

11

S. P. Redmond (2007). On zero-divisor graphs of small finite commutative rings. Discrete Math.. 307, 1155-1166

12

C. Wickham (2008). Classification of Rings with Genus One Zero-Divisor Graphs. Comm. Algebra. 36, 325-345

13

A. T. White (1973). Graphs, Groups and Surfaces. North-Holland, Amsterdam.