The study of random geometric graphs begins with Gilbert (1961) in his paper titled as "Random Plane Networks" published in Journal of the Society for Industrial Applied Mathematic. In this thesis, we study the RGG, whose vertices have the densities with unbounded support. We study the various properties of RGG and are interested in both exact and asymptotic results for one-dimensional as well as d-dimensional (d > 1). % The thesis is divided in four chapters. Chapter 1 introduces the concept and the utility of RGG and gives an idea about the techniques and tool which are used in the thesis. % In chapter 2 we study the one dimensional random geometric (random interval) graph when the location of the nodes are independent and exponentially distributed. We derive exact results and limit theorems for the connectivity and other properties associated with this random graph. We show that the asymptotic properties of a graph with a truncated exponential distribution can be obtained using the exponential random geometric graph. % In chapter 3 we prove the criticality of the exponential rate of decay for the largest nearest neighbor link in RGG. %

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