Nonadaptive group testing involves grouping arbitrary subsets of n items into different pools. Each pool is then tested and defective items are identified. A fundamental question involves minimizing the number of pools required to identify at most d defective items. Motivated by applications in network tomography, sensor networks and infection propagation, a variation of group testing problems on graphs is formulated. Unlike conventional group testing problems, each group here must conform to the constraints imposed by a graph. For instance, items can be associated with vertices and each pool is any set of nodes that must be path connected. In this paper, a test is associated with a random walk. In this context, conventional group testing corresponds to the special case of a complete graph on n vertices. For interesting classes of graphs a rather surprising result is obtained, namely, that the number of tests required to identify d defective items is substantially similar to what is required in conventional group testing problems, where no such constraints on pooling is imposed. Specifically, if T(n) corresponds to the mixing time of the graph G, it is shown that with $m = O(d^2 T^2 (n) log(n/d))$ nonadaptive tests, one can identify the defective items. Consequently, for the Erdos-Rényi random graph G(n, p), as well as expander graphs with constant spectral gap, it follows that $m = O(d^2 log^3 n)$ non-adaptive tests are sufficient to identify d defective items. Next, a specific scenario is considered that arises in network tomography, for which it is shown that m = O(d 3 log 3 n) nonadaptive tests are sufficient to identify d defective items. Noisy counterparts of the graph constrained group testing problem are considered, for which parallel results are developed. We also briefly discuss extensions to compressive sensing on graphs.