Random graphs are encountered in a number of disciplines: pure and applied mathematics, biology, computer science, engineering, etc. Several models have been considered and a number of questions have been answered for each model. This research investigates establishment of a unifying framework within which all these models can be treated. The framework proposed is that of Generalized Euclidean Random Graphs in which a link's probability of existence is a function of the Euclidean distance between the nodes which the link connects. The research considers a number of questions related to the probabilistic behavior of such graphs, such as connectivity, network diameter, Euclidean length of links, length of minimum path lengths, etc. Whenever possible, analytical methods are used to find exact solutions. When analysis falls short, bounding and/or approximation techniques is used, verified by computer simulations, to obtain bounds and/or approximate answers. The impact of this research should be significant. As the vast literature on graph theory suggests, graph theory has penetrated a large number of scientific and engineering disciplines. We are only now beginning to understand the behavior of random graphs. The additional structure that the distance imposes will allow an entire collection of new problems to be modeled by these graphs. Therefore, besides the theoretical interest, this research promises to be of great interest to a very diverse set of scientists and engineers that will be able to use the results of the research, as well as stimulate new areas of research.
