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In this paper, we study the recovery of the graph topology or structure. We first extend our previous work on graph signal recovery to present a joint graph signal and structure recovery framework. By doing this, we allow the algorithm to learn a graph structure from noisy and incomplete graph signals and recover the graph signals at the same time. In this paper, we particularly focus on the specific subproblem of graph structure learning and develop algorithms towards this problem and analyze them. We briefly study the implications when the underlying true graph structure is irregular or regular. Finally, we validate the proposed methods for both synthetic data and the real-world recovery problem of semi-supervised digit-image classification.