Abstract
In this paper, we study the topological graph theoretic properties such as planarity, toroidality and bi-toroidality of the total dot product graph of a commutative ring. In particular, we characterize an isomorphism class of commutative rings R for which TD(R) has genus one or two. This leads to the characterization of all commutative rings whose ZD(R) has genus one or two. It is shown that for any commutative ring R, TD(R) is a bi-toroidal graph if and only if R is ring isomorphic to \(\frac{{\mathbb {Z}}_2\left[ x\right] }{\left\langle x^2+x+1\right\rangle }\times \frac{{\mathbb {Z}}_2\left[ x\right] }{\left\langle x^2+x+1\right\rangle }\) and \({\mathbb {Z}}_5\times {\mathbb {Z}}_5.\)
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Acknowledgements
The research work of the second author is supported by the Dr. D.S. Kothari Postdoctoral Fellowship (No.F.4-2/2006 (BSR)/MA/ 17-18/ 0045), University Grants Commission, Government of India.
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Communicated by Rahul Roy.
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Selvakumar, K., Ramanathan, V. & Selvaraj, C. On the genus of dot product graph of a commutative ring. Indian J Pure Appl Math 54, 558–567 (2023). https://doi.org/10.1007/s13226-022-00275-0
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DOI: https://doi.org/10.1007/s13226-022-00275-0