On Planarity and the Metric Dimension of Proper Divisor Graph of Positive Integers

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.