%0 Journal Article
%T A typical graph structure of a ring
%J Transactions on Combinatorics
%I University of Isfahan
%Z 2251-8657
%A Kala, R.
%A Kavitha, S.
%D 2015
%\ 06/01/2015
%V 4
%N 2
%P 37-44
%! A typical graph structure of a ring
%K local ring
%K nilpotent
%K planar
%K Artinian ring
%R 10.22108/toc.2015.6177
%X 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 $xyneq 0$, where $mathcal{Z}_N(R)={xin R: xy~text{is nilpotent, for some} yin R^*}$. 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 $Gamma_N^*(R)$.
%U https://toc.ui.ac.ir/article_6177_95c66f3ddffbab1d427f14e4b0d0e823.pdf