Edge bundling reduces the visual complexity of drawings of large and complex graphs by clustering "compatible" edges. However, it often introduces distortion by bundling "unrelated" edges, resulting in misleading, ambiguous drawings. Moreover, existing edge bundling methods often have high computational complexity. We present new edge bundling methods and faithfulness metrics for edge bundling using spectral sparsification, which sparsifies a graph G into a subgraph G' with O(n log n) edges, based on the effective resistance values of edges, preserving the spectrum of G, closely related to important structural properties of G, such as connectivity, clustering, and the commute distance. We first present a new edge bundling method SEB (Spectral Edge Bundling), introducing effective resistance-based compatibility for spectral-faithful bundling, improving distortion and ambiguity. Then, we present a general framework FEB (Fast Edge Bundling) utilizing spectral sparsification to improve the efficiency of existing bundling methods while maintaining a similar quality. We also present FBQ (Fast Bundling Quality) framework for proxy bundle faithfulness metrics, for measuring how FEB faithfully preserves the ground truth structure in the original edge bundling, with two variants, FBQ_JS (utilizing Jaccard Similarity) and FBQ_SQ (utilizing sampling quality metrics). Extensive experiments using various real-world and synthetic graphs demonstrate the effectiveness of SEB for edge bundling, outperforming state-of-art bundling methods on quality metrics, with 46% and 17% average improvement in distortion and ambiguity respectively for SEB2. Furthermore, experiments successfully demonstrate the efficiency of the FEB framework, with 61% runtime improvement over the original edge bundling methods without sparsification, while maintaining a similar quality, with 74% similarity based on FBQ_SQ.