Extended total graph associated to finite commutative rings

Aaqib Altaf; S. Pirzada; Ahmad M. Alghamdi; Eman S. Almotairi

doi:10.3842/umzh.v76i5.7494

Aaqib Altaf Department of Mathematics, University of Kashmir, Srinagar, India
S. Pirzada Department of Mathematics, University of Kashmir, Srinagar, India
Ahmad M. Alghamdi Mathematics Department, Faculty of Sciences, Umm Al-Qura University, Makkah, Saudi Arabia
Eman S. Almotairi Department of Mathematics, College of Sciences, Qassim University, Buraydah, Saudi Arabia

DOI: https://doi.org/10.3842/umzh.v76i5.7494

Keywords: Finite commutative rings, Total graph, Clique number

Abstract

UDC 512.5

For a commutative ring $R$ with nonzero identity $1\neq 0$, let $Z(R)$ denote the set of zero divisors. The total graph of $R$ denoted by $T_{\Gamma}(R)$ is a simple graph in which all elements of $R$ are vertices and any two distinct vertices $x$ and $y$ are adjacent if and only if $x+y\in Z(R)$. In this paper, we define an extension of the total graph denoted by $T(\Gamma^{e}(R))$ with vertex set as $Z(R),$ and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y\in Z^*(R)$, where $ Z^{*}(R)$ is the set of nonzero zero divisors of $R$. Our main aim is to characterize the finite commutative rings whose $T(\Gamma^{e}(R))$ has clique numbers $1,2,$ and $3$. In addition, we characterize finite commutative nonlocal rings $R$ for which the corresponding graph $T(\Gamma^{e}(R))$ has the clique number $4.$

References

D. F. Anderson, Ayman Badawi, The total graph of a commutative ring, J. Algebra, 320, № 7, 706–2719 (2008). DOI: https://doi.org/10.1016/j.jalgebra.2008.06.028

D. F. Anderson, P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217, 434–447 (1999). DOI: https://doi.org/10.1006/jabr.1998.7840

T. Asir, T. Tamizh Chelvam, On the toal graph and its complement of a commutative ring, Comm. Algebra, 41, № 10, 3820–3835 (2013). DOI: https://doi.org/10.1080/00927872.2012.678956

I. Beck, Coloring of a commutative rings, J. Algebra, 116, 208–226 (1988). DOI: https://doi.org/10.1016/0021-8693(88)90202-5

D. Bennis, J. Mikram, F. Taraza, On the extended zero divisor graph of commutative rings, Turkish J. Math., 40, № 2, Article 12 (2016). DOI: https://doi.org/10.3906/mat-1504-61

A. Cherrabi, H. Essannouni, E. Jabbouri, A. Ouadfel, On a new extension of the zero-divisor graph, Algebra Colloq., 27, № 3, 469–476 (2020). DOI: https://doi.org/10.1142/S1005386720000383

A. Cherrabi, H. Essannouni, E. Jabbouri, A. Ouadfel, On a new extension of the zero-divisor graph (II), Beitr Algebra und Geom., 62, 945–953 (2021). DOI: https://doi.org/10.1007/s13366-020-00559-8

D. S. Dummit, R. M. Foote, Abstract algebra, vol.~3, Wiley Hoboken (2004).

Qiong Liu, Tongsuo Wu, Ji. Guo, Finite rings whose graphs have clique number less than five, Algebra Colloq., 28, № 3, 533–540 (2021). DOI: https://doi.org/10.1142/S1005386721000419

Qiong Liu, Tongsuo Wu, Ji. Guo, On finite local rings with clique number four, Algebra Colloq., 29, № 1, 23–38 (2022). DOI: https://doi.org/10.1142/S1005386722000037

I. Kaplansky, Commutative rings, Univ. Chicago Press, Chicago, London (1974).

S. Pirzada, An introduction to graph theory, Univ. Press, Orient Blakswan, Hyderabad (2012).

S. Pirzada, A. Altaf, Cliques in the extended zero-divisor graph of finite commutative rings, Commun. Combin. and Optim. (to appear); https://doi.org/10.22049/cco.2023.28740.1693.

S. Pirzada, A. Altaf, Co-unit graphs associated to ring of integers modulo $n$, Acta Univ. Sapientiae Math., 14, № 2, 98–106 (2022). DOI: https://doi.org/10.2478/ausm-2022-0020

S. Pirzada, A. Ahmad, On graphs with same metric and upper dimension, Discrete Math., Algorithms and Appl., 13, № 2, Article ID 2150015 (2021). DOI: https://doi.org/10.1142/S1793830921500154

S. P. Redmond, On zero-divisor graphs of small finite commutative rings, Discrete Math., 307, 1155–1166 (2007).

S. P. Redmond, Corrigendum to ``On zero-divisor graphs of small finite commutative rings'', Discrete Math., 307, 1155–1166 (2007). DOI: https://doi.org/10.1016/j.disc.2006.07.025

Yanzhao Tian, Lixiang Li, Comments on the clique number of zero-divisor graphs of $Z_{n}$, J. Math., Article ID 6591317 (2022). DOI: https://doi.org/10.1155/2022/6591317