The modified divisor graph of commutative ring
Abstract
Let \( R \) be a finite commutative ring with unity. We have introduced a divisor graph of the ring, denoted by \( D[R] \). It is an undirected simple graph with a vertex set \( V = R - \{0, 1\} \), and two distinct vertices \( u \) and \( v \) in \( V \) are adjacent if and only if there exists \( w \in R \) such that \( v = uw \) or \( u = vw \). Clearly, \( 0 \) and unit elements are adjacent to all elements of the ring. Thus, we adopt the idea of Anderson and Livingston and remove the zero and unit vertices from the vertex set. A new graph is called the modified divisor graph, noted by symbol \( D_0[R] \).In this study, we have focused on the structural properties of \( D_0[\mathbb{Z}_n] \). Moreover, we have determined the diameter, girth, clique number, vertex connectivity, and edge connectivity of \( D_0[\mathbb{Z}_n] \) for every natural number \( n \). The purpose of studying this graph is to find relationships between the ring theoretic properties of \( R \) and the graph theoretic properties of \( D_0[R] \).
Keywords
Full Text:
PDFReferences
S. Akbari and Mohammadian. "On the zero-divisor graph of a commutative ring." *Journal of Algebra*, 274(2):847–855, 2004.
I. Beck. "Colouring of a commutative ring." *Journal of Algebra*, 116:208–226, 1998.
N. L. B. S. Reddy, R. S. Jain. "Eulerian of zero divisor graph ( gamma(z_n) )." *arXiv preprint arXiv*, 2020.
D. F. Anderson and P. S. Livingston. "The zero-divisor graph of a commutative ring." *Journal of Algebra*, 217:434–447, 1999.
A. Das. "Subspace inclusion graph of a vector space." *Communications in Algebra*, 44:4724–4731, 2016.
Khandare, Jogdand, and Muneshwar. "On the divisor graph of finite commutative ring." *J. Math. Comput. Sci.*, 12:37–41, 2022.
M. K. M. S. Rahman. "On hamiltonian cycles and hamiltonian paths." *Information Processing Letters*, 49:37–41, 2005.
P. Mathil, B. Baloda, and J. Kumar. "On the cozero-divisor graphs associated to rings." *AKCE International Journal of Graphs and Combinatorics*, 19(3):238–248, 2022.
K. B. R. A. Muneshwar. "Some properties of open subset intersection graph of a topological space." *Journal of Information and Optimization Sciences*, 22:1007–1018, 2021.
Redmond and S. P. "The zero-divisor graph of a non-commutative ring." *International J. Commutative Rings*, 1:203–211, 2002.
S. P. Redmond. "An ideal-based zero-divisor graph of a commutative ring." *Communications in Algebra*, 31(9):4425–4443, 2003.
A. M. R. M. S. Akabari, M. Hababi. "On the inclusion graph of ring." *Electronic notes in discrete math*, 45:73–78, 2014.
D. West. *Introduction to graph theory.* Prentice Hall, 2001.
DOI: http://dx.doi.org/10.23755/rm.v51i0.1420
Refbacks
- There are currently no refbacks.
Copyright (c) 2024 prakash Dnyanba khandare, Suryakant M Jogdand, Rajesh A Muneshwar
This work is licensed under a Creative Commons Attribution 4.0 International License.
Ratio Mathematica - Journal of Mathematics, Statistics, and Applications. ISSN 1592-7415; e-ISSN 2282-8214.