|
Algebra and Discrete Mathematics, 2017, Volume 24, Issue 2, Pages 181–190
(Mi adm626)
|
|
|
|
RESEARCH ARTICLE
Some properties of the nilradical and non-nilradical graphs over finite commutative ring $\mathbb{Z}_n$
Shalini Chandraa, Om Prakashb, Sheela Suthara a Department of Mathematics and Statistics, Banasthali Vidyapith, Banasthali, Rajasthan 304022, India
b Department of Mathematics, IIT Patna, Patliputra colony, Patna 800013, India
Abstract:
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.
Received: 24.09.2015 Revised: 25.02.2016
Citation:
Shalini Chandra, Om Prakash, Sheela Suthar, “Some properties of the nilradical and non-nilradical graphs over finite commutative ring $\mathbb{Z}_n$”, Algebra Discrete Math., 24:2 (2017), 181–190
Linking options:
https://www.mathnet.ru/eng/adm626 https://www.mathnet.ru/eng/adm/v24/i2/p181
|
Statistics & downloads: |
Abstract page: | 259 | Full-text PDF : | 332 | References: | 21 |
|