Summary: Spectral measures have long been used to quantify the robustness of real-world graphs. For example, spectral radius (or the principal eigenvalue) is related to the effective spreading rates of dynamic processes (e.g., rumor, disease, information propagation) on graphs. Algebraic connectivity (or the Fiedler value), which is a lower bound on the node and edge connectivity of a graph, captures the “partitionability” of a graph into disjoint components. In this work we address the problem of modifying a given graph’s structure under a given budget so as to maximally improve its robustness, as quantified by spectral measures. We focus on modifications based on degree-preserving edge rewiring, such that the expected load (e.g., airport flight capacity) or physical/hardware requirement (e.g., count of ISP router traffic switches) of nodes remain unchanged. Different from a vast literature of measure-independent heuristic approaches, we propose an algorithm, called EdgeRewire, which optimizes a specific measure of interest directly. Notably, EdgeRewire is general to accommodate six different spectral measures. Experiments on real-world datasets from three different domains (Internet AS-level, P2P, and airport flights graphs) show the effectiveness of our approach, where EdgeRewire produces graphs with both (i) higher robustness, and (ii) higher attack-tolerance over several state-of-the-art methods.