The availability of Graphics Processing Units (GPUs) with multicore architecture have enabled parallel computations using extensive multi-threading. Recent advancements in computer hardware have led to the usage of graphics processors for solving general purpose problems. Using GPUs for computation is a highly efficient and low-cost alternative as compared to currently available multicore Central Processing Units (CPUs). Also, in the past decade there has been tremendous growth in the World Wide Web and Online Social Networks. Social networking sites such as Facebook, Twitter and LinkedIn, with millions of users are a huge source of data. These data sets can be used for research in the fields of anthropology, social psychology, economics among others. Our research focuses on converting real-world problems into graph theoretic problems and using GPUs to solve them. The graph problems that we focus on in our research involve counting the number of subgraphs that satisfy a given property. For example, given a graph G=(V,E) and an integer k<=|V|, we provide algorithms to count the number of: a) connected subgraphs of size k; b) cliques of size k; and c) independent sets of size k, and other similar problems. Also, properties that are affected by the dynamic nature of the graphs i.e., addition or removal of edges or nodes, for example change in the number of triangles and connected components in the graph, are also studied. Sequential access to global memory and contention at the size-limited shared memory have been main impediments to fully exploiting potential performance in GPUs. Therefore, we propose novel memory storage and retrieval methods, based on using search techniques on graphs and converting it into trees, that enable parallel graph computations to overcome the above issues. We also analyze and utilize primitives such as memory access coalescing and avoiding partition camping that offset the increase in access latency of using a slower but larger global memory. In addition, we introduce graph compression techniques that further reduce memory requirements and overheads. Our experimental results for the GPU implementation show a significant speedup over the CPU counterpart for the problems described above.