Simultaneous localization and mapping (SLAM) is a critical capability in autonomous navigation, but memory and computational limits make long-term application of common SLAM techniques impractical; a robot must be able to determine what information should be retained and what can safely be forgotten. In graph-based SLAM, the number of edges (measurements) in a pose graph determines both the memory requirements of storing a robot's observations and the computational expense of algorithms deployed for performing state estimation using those observations, both of which can grow unbounded during long-term navigation. Motivated by these challenges, we propose a new general purpose approach to sparsify graphs in a manner that maximizes algebraic connectivity, a key spectral property of graphs which has been shown to control the estimation error of pose graph SLAM solutions. Our algorithm, MAC (for maximizing algebraic connectivity), is simple and computationally inexpensive, and admits formal post hoc performance guarantees on the quality of the solution that it provides. In application to the problem of pose-graph SLAM, we show on several benchmark datasets that our approach quickly produces high-quality sparsification results which retain the connectivity of the graph and, in turn, the quality of corresponding SLAM solutions.