We examine the problem of detecting negative cycles in a dynamic graph, which is a fundamental problem that arises in electronic design automation and systems theory. We introduce the concept of adaptive negative cycle detection, in which a graph changes over time, and negative cycle detection needs to be done periodically, but not necessarily after every individual change. Such scenarios arise, for example, during iterative design space exploration for hardware and software synthesis. We present an algorithm for this problem, called the Adaptive Bellman-Ford (ABF) algorithm, based on a novel extension of the well known Bellman-Ford algorithm. The ABF algorithm allows us to systematically adapt information for a given graph to a modified version of the graph. We show that the ABF algorithm significantly outperforms previously available approaches for dynamic graphs, which either recompute negative cycle information from scratch whenever a graph is modified, or process the modifications one at a time ("incrementally"). As an application of the ABF technique, we show that it can be used to obtain a very fast implementation of Lawler's technique for the computation of the maximum-cycle mean (MCM) of a graph, especially for a certain important kind of sparse graph. We further illustrate the application of the ABF technique to design-space exploration by developing automated search techniques for scheduling iterative data-flow graphs.