Geodesic
Some properties of the nilradical and non-nilradical graphs over finite commutative ring~$\mathbb{Z}_n$
Algebra and discrete mathematics, Tome 24 (2017) no. 2, pp. 181-190

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Let $\mathbb{Z}_n$ be the finite commutative ring of residue classes modulo $n$ with identity and $\Gamma(\mathbb{Z}_n)$ be its zero-divisor graph. In this paper, we investigate some properties of nilradical graph, denoted by $N(\mathbb{Z}_n)$ and non-nilradical graph, denoted by $\Omega(\mathbb{Z}_n)$ of $\Gamma(\mathbb{Z}_n)$. In particular, we determine the Chromatic number and Energy of $N(\mathbb{Z}_n)$ and $\Omega(\mathbb{Z}_n)$ for a positive integer $n$. In addition, we have found the conditions in which $N(\mathbb{Z}_n)$ and $\Omega(\mathbb{Z}_n)$ graphs are planar. We have also given MATLAB coding of our calculations.
Keywords: commutative ring, zero-divisor graph, nilradical graph, non-nilradical graph, chromatic number, planar graph, energy of a graph. 
@article{ADM_2017_24_2_a0,
     author = {Shalini Chandra and Om Prakash and Sheela Suthar},
     title = {Some properties of the nilradical and non-nilradical graphs over finite commutative ring~$\mathbb{Z}_n$},
     journal = {Algebra and discrete mathematics},
     pages = {181--190},
     publisher = {mathdoc},
     volume = {24},
     number = {2},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2017_24_2_a0/}
}
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Shalini Chandra; Om Prakash; Sheela Suthar. Some properties of the nilradical and non-nilradical graphs over finite commutative ring~$\mathbb{Z}_n$. Algebra and discrete mathematics, Tome 24 (2017) no. 2, pp. 181-190. http://geodesic.mathdoc.fr/item/ADM_2017_24_2_a0/