Abstract
Consider the random geometric graph \(G = G(n,r_n,f)\) consisting of n nodes independently distributed in \(S = \left[ -\frac{1}{2},\frac{1}{2}\right] ^2\) each according to a density \(f({\cdot })\). Two nodes u and v are joined by an edge if the Euclidean distance between them is less than \(r_n.\) Let \(C_G\) denote the component of G containing the largest number of nodes and denote \(\text {diam}(C_G)\) to be its diameter. Let s and t be the nodes of G closest to \(\left( -\frac{1}{2},\frac{1}{2}\right) \) and \(\left( \frac{1}{2},\frac{1}{2}\right) ,\) respectively and let \(d_G(s,t)\) denote their graph distance. We prove that the normalized diameter \(\frac{r_n}{\sqrt{2}} \text {diam}(C_G)\) and the stretch \(r_nd_G(s,t)\) both converge to one in probability if \(nr_n^2 \rightarrow \infty \) as \(n \rightarrow \infty \).
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Acknowledgements
I thank Professors Rahul Roy, Thomas Mountford, Federico Camia and the referees for crucial comments that led to an improvement of the paper. I also thank Professors Rahul Roy, Thomas Mountford, Federico Camia and the National Board for Higher Mathematics for my fellowships.
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Ganesan, G. Stretch and Diameter in Random Geometric Graphs. Algorithmica 80, 300–330 (2018). https://doi.org/10.1007/s00453-016-0253-5
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DOI: https://doi.org/10.1007/s00453-016-0253-5