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Total graph of a lattice

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Abstract

In this paper, we prove that the study of the subgraph \(T(Z^*(L))\) of the total graph T(L) of a lattice L is essentially the study of the zero-divisor graph of a poset. Also, we prove that the graph \(T^c(Z^*(L))\) is weakly perfect whereas \(T(Z^*(L))\) is not weakly perfect. The graph \(T(Z^*(L))\) and its complement \(T^c(Z^*(L))\) are shown to be a perfect graph if and only if L has at most four atoms. In the concluding section, we establish that, in the context of a commutative reduced ring R, the total graph, the annihilating ideal graph, the complement of the co-annihilating ideal graph, and the complement of the comaximal ideal graph coincide.

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Acknowledgements

The authors are thankful to the referee for his/her suggestions that improved the presentation of the paper. The first author is grateful to the Department of Mathematics, Savitribai Phule Pune University, Pune, for awarding Professor T. T. Raghunathan fellowship during his Master’s Degree (2010–2012). Also, the authors are thankful to Mr. Nilesh Khandekar for his valuable suggestions and fruitful discussion during the completion of this paper. Both authors contributed equally to the study of the total graph of a lattice. Both authors read and approved the final version of the manuscript.

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Authors and Affiliations

  1. GES’s Shri Bhausaheb Vartak Arts, Commerce and Science College, Borivali, Maharashtra, 400091, India

    Pravin Gadge

  2. Department of Mathematics, Savitribai Phule Pune University, Pune, Maharashtra, 411007, India

    Vinayak Joshi

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  1. Pravin Gadge
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  2. Vinayak Joshi
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Both authors contributed equally to the study of the total graph of a lattice. Both authors read and approved the final version of the manuscript.

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Correspondence to Vinayak Joshi.

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The authors declare that there is no conflict of interests regarding the publishing of this paper.

Additional information

Communicated by Shariefuddin Pirzada.

Dedicated to Late Professor T. T. Raghunathan.

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Gadge, P., Joshi, V. Total graph of a lattice. Indian J Pure Appl Math (2024). https://doi.org/10.1007/s13226-024-00551-1

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