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.

Voir la notice de l'article provenant de la source Math-Net.Ru

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/}
}
TY  - JOUR
AU  - Shalini Chandra
AU  - Om Prakash
AU  - Sheela Suthar
TI  - Some properties of the nilradical and non-nilradical graphs over finite commutative ring~$\mathbb{Z}_n$
JO  - Algebra and discrete mathematics
PY  - 2017
SP  - 181
EP  - 190
VL  - 24
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ADM_2017_24_2_a0/
LA  - en
ID  - ADM_2017_24_2_a0
ER  - 
%0 Journal Article
%A Shalini Chandra
%A Om Prakash
%A Sheela Suthar
%T Some properties of the nilradical and non-nilradical graphs over finite commutative ring~$\mathbb{Z}_n$
%J Algebra and discrete mathematics
%D 2017
%P 181-190
%V 24
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ADM_2017_24_2_a0/
%G en
%F ADM_2017_24_2_a0
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/

[1] M. R. Ahmadi and R. J. Nezhad, “Energy and Wiener Index of Zero Divisor Graphs”, Iranian J. Math. Chem., 2:1 (2011), 45–51 | Zbl

[2] D. F. Anderson and P. S. Livingston, “The zero-divisor graph of a commutative ring”, J. Algebra, 217 (1999), 434–447 | DOI | MR | Zbl

[3] I. Beck, “Coloring of Commutative rings”, J. Algebra, 116 (1988), 208–226 | DOI | MR | Zbl

[4] R. Belshoff and J. Chapman, “Planar zero divisor graphs”, J. Algebra, 316:1 (2007), 471–480 | DOI | MR | Zbl

[5] V. K. Bhat, Ravi Raina, Neeraj Nehra and Om Prakash, “A Note on zero divisor graph over Rings”, Int J. Contemp. Math. Sci., 2:14 (2007), 667–671 | DOI | MR | Zbl

[6] A. Bishop, T. Cuchta, K. Lokken and O. Pechenik, “The Nilradical and Non-Nilradical Graphs of Commutative Rings”, Int. J. Algebra, 2:20 (2008), 981–994 | MR | Zbl

[7] F. Harary, Graph Theory, 1st ed., Addison-Wesley, Boston, 1969 | MR | Zbl

[8] C. Musli, Introduction to Ring and Modules, 2nd ed., Narosa Publishing House, New Delhi, 1997