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Visual dialog is a task of answering a sequence of questions grounded in an image using the previous dialog history as context. In this paper, we study how to address two fundamental challenges for this task: (1) reasoning over underlying semantic structures among dialog rounds and (2) identifying several appropriate answers to the given question. To address these challenges, we propose a Sparse Graph Learning (SGL) method to formulate visual dialog as a graph structure learning task. SGL infers inherently sparse dialog structures by incorporating binary and score edges and leveraging a new structural loss function. Next, we introduce a Knowledge Transfer (KT) method that extracts the answer predictions from the teacher model and uses them as pseudo labels. We propose KT to remedy the shortcomings of single ground-truth labels, which severely limit the ability of a model to obtain multiple reasonable answers. As a result, our proposed model significantly improves reasoning capability compared to baseline methods and outperforms the state-of-the-art approaches on the VisDial v1.0 dataset. The source code is available at https://github.com/gicheonkang/SGLKT-VisDial.