ON THE NILPOTENT DOT PRODUCT GRAPH OF A COMMUTATIVE RING

Articles in Press

Document Type : Original Manuscript

Authors

Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India.

Abstract

Let $\mathscr{B}$ be a commutative ring with $1\neq 0$, $1\leq m<\infty$ be an integer and $\mathcal{R}=\mathscr{B}\times \mathscr{B}\times \cdot \cdot \cdot \times \mathscr{B}$ ($m$ times). In this paper, we introduce two types of (undirected) graphs, total nilpotent dot product graph denoted by $\mathcal{T_{N}D(\mathcal{R})}$ and nilpotent dot product graph denoted by $\mathcal{Z_ND(\mathcal{R})}$, in which vertices are from $\mathcal{R}^\ast = \mathcal{R}\setminus \{(0,0,...,0)\}$ and $\mathcal{Z_{N}(\mathcal{R})}^*$ respectively, where $\mathcal{Z_{N}(\mathcal{R})}^{*}=\{w\in \mathcal{R}^*| wz\in \mathcal{N(R)}, \mbox{for some }z\in \mathcal{R}^*\} $. Two distinct vertices $w=(w_1,w_2,...,w_m)$ and $z=(z_1,z_2,...,z_m)$ are said to be adjacent if and only if $w\cdot z\in \mathcal{N}(\mathscr{B})$ (where $w\cdot z=w_1z_1+\cdots+w_mz_m$, denotes the normal dot product and $\mathcal{N}(\mathscr{B})$ is the set of nilpotent elements of $\mathscr{B}$). We study about connectedness, diameter and girth of the graphs $\mathcal{T_ND(R)}$ and $\mathcal{Z_ND(R)}$. Finally, we establish the relationship between $\mathcal{T_ND(R)}$, $\mathcal{Z_ND(R)}$, $\mathcal{TD(R)}$ and $\mathcal{ZD(R)}$.

Keywords

References

  1. S. Akbari and A. Mohammadian, On the zero-divisor graph of a commutative ring, J.

Algebra, 274(2) (2004), 847-855.

  1. S. Akbari and A. Mohammadian, Zero-divisor graph of a non-commutative rings, J. Al

gebra, 296(2) (2006), 462-479.

  1. D. F. Anderson, T. Asir, A. Badawi and T. Tamizh Chelvam, Graphs from rings, First

ed., Springer Nature Switzerland AG, 2021.

  1. D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra, 320(7)

(2008), 2706–2719.

  1. D. F. Anderson, R. Levy and J. Shapirob, Zero-divisor graphs, von Neumann regular rings

and Boolean algebras, J. Pure Appl. Algebra, 180(3) (2003), 221–241.

  1. D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J.

Algebra, 217 (1999), 434–447.

  1. D. D. Anderson and M. Naseer, Beck’s coloring of a commutative ring, J. Algebra, 159(2)

(1993), 500–514.

  1. M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley

Publishing Company, Massachusetts, London, Ontario, 1969.

  1. A. Badawi, On the dot product graph of a commutative ring, Comm. Algebra, 43(1)

(2015), 43–50.

  1. I. Beck, Coloring of commutative rings, J. Algebra, 116(1) (1988), 208–226.
  2. P. W. Chen, A kind of graph structure of rings, Algebra Colloq., 10 (2003), 229–238.
  3. F. DeMeyer, T. McKenzia and K. Schneider, The zero-divisor graph of a commutative

semigroup, Semigroup Fourm, 65 (2002), 206–214.

  1. R. Kala and S. Kavitha, A typical graph structure of a ring, Transactions on Combina

torics, 4(2) (2015), 37–44.

  1. R. Kala and S. Kavitha, Nilpotent graph of genus one, Discrete Math. Algorithms Appl.,

6 (2014), Article ID: 1450037.

  1. J. D. LaGrange, Complemented zero divisor graphs and Boolean rings, J. Algebra, 315(2)

(2007), 600–611.

 

Journal of Algebraic Systems

Articles in Press, Accepted Manuscript
Available Online from 09 April 2024

Files

Share

How to cite

Statistics