Abstract
The proper divisor graph \(\Upsilon _n\) of a positive integer n is an induced subgraph of the zero divisor graph \(\Gamma (\mathbb {Z}_n)\) of the ring \(\mathbb {Z}_n\) and also plays an important role in studying the Laplacian spectrum of \(\Gamma (\mathbb {Z}_n)\). The vertices of \(\Upsilon _n\) are the proper divisors of n and two distinct vertices x, y are adjacent if and only if n divides xy. We determine the values of n for which \(\Upsilon _n\) is split, co-graph, planar, outer-planar and ring graph. We also determine the decycling number, metric dimension, strong metric dimension and fixing number of the zero divisor graph of a monogenic semigroup.
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Acknowledgements
The first author is thankful to the University Grants Commission, India, for financial support as UGC-SRF (Award. no. 11-04-2016-421922). Also the second author is supported from MHRD, Govt of India, in the form of Senior Research Fellowship.
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Khanra, B., Ghosh, B.D., Das, S. et al. On Planarity and the Metric Dimension of Proper Divisor Graph of Positive Integers. Natl. Acad. Sci. Lett. (2023). https://doi.org/10.1007/s40009-023-01362-4
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DOI: https://doi.org/10.1007/s40009-023-01362-4