A Note on the Double Total Graph \(T_u(\Gamma(R))\) and \(T_u(\Gamma(\mathbb{Z}_n \times \mathbb{Z}_m))\)
DOI:
https://doi.org/10.26713/cma.v12i1.1466Keywords:
Fusible ring, Weakly unit fusible ring, Unit graph, Total graph, Double total graphAbstract
Considering a commutative ring \(R\) with unity as the set of vertices and two vertices \(x\) and \(y\) are adjacent if and only if \(u+(x+y) \in Z(R)\) for some \(u \in U(R)\), the resulting graph \(T_{u}(\Gamma(R))\) is known as the double total graph. In this paper we find the degree of any vertex in \(T_{u}(\Gamma(R))\) for a weakly unit fusible ring \(R\) and domination number of \(T_{u}(\Gamma(R))\) for any ring \(R\). Also, we investigate the properties of \(T_{u}(\Gamma(\mathbb{Z}_{n}\times\mathbb{Z}_{m}))\) and characterize $R$ in terms of toroidal \(T_{u}(\Gamma(R))\).
Downloads
References
D. F. Anderson and A. Badawi, The total graph of a commutative ring, Journal of Algebra 320 (2008), 2706 – 2719, DOI: 10.1016/j.jalgebra.2008.06.028.
N. Ashrafi, H. R. Maimani, M. R. Pournaki and S. Yassemi, Unit graphs associated with rings, Communications in Algebra 38(8) (2010), 2851 – 2871, DOI: 10.1080/00927870903095574.
A. M. Dhorajia, Total graph of the ring Zn £Zm, Discrete Mathematics, Algorithms and Applications 7(1) (2015), 1550004, DOI: 10.1142/S1793830915500044.
E. Ghashghaei and W. Wm. McGovern, Fusible rings, Communications in Algebra 45(3) (2017), 1151 – 1165, DOI: 10.1080/00927872.2016.1206347.
M. C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, San Diego (1980).
N. R. Singh and S. Dutta, The double total graph of a commutative ring, Advances and Applications in Discrete Mathematics 24(2) (2020), 99 – 115, DOI: 10.17654/DM024020099.
H. Su, K. Noguchi and Y. Zhou, Finite commutative rings with higher genus unit graphs, Journal of Algebra and Its Applications 14(1) (2015), 1550002, DOI: 10.1142/S0219498815500024.
D. B. West, Introduction to Graph Theory, 2nd edition, Prentice Hall (2001).
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.
