A Note on the Double Total Graph \(T_u(\Gamma(R))\) and \(T_u(\Gamma(\mathbb{Z}_n \times \mathbb{Z}_m))\)

Ngangom Rojitkumar Singh, Sanghita Dutta

Abstract


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 \(T_{u}(\Gamma(R))\).


Keywords


Fusible ring; Weakly unit fusible ring; Unit graph; Total graph; Double total graph

Full Text:

PDF

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).




DOI: http://dx.doi.org/10.26713%2Fcma.v12i1.1466

Refbacks

  • There are currently no refbacks.


eISSN 0975-8607; pISSN 0976-5905