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The graph is a data structure that exists widely around us, including traditional fields like physics, biology, and cosmology, as well as emergent fields like social networks, software engineering, and financial trading platforms. The graph-structured data contains objects (nodes) information and reflects their relationships (edges). The learning tasks become more challenging when considering the nodes and edge information simultaneously. Traditional machine learning methods focus on nodes' attributes but ignore the structural information. We are now in an era of deep learning, which outperforms traditional machine learning methods in a wide range of tasks and has a significant impact on our daily lives. Driving by deep learning and neural networks, the deep learning-based graph neural networks (GNNs) become convincing and attractive tools to handle this non-Euclidean data structure. The dissertation thesis includes my research works throughout the Ph.D. research in two directions of graph data mining. The first direction is about the innovation and improvement of graph neural networks. A large number of GNNs have appeared, but as a general representation learning model, there are still some difficult topics worth delving into. I focus on three questions: Unsupervised/self-supervised Learning of GNNs, GNNs for heterogeneous graphs, and Training larger and deeper GNNs. Concerning unsupervised/self-supervised learning of GNNs, the dissertation introduces my research works contributing to it in Chapter 3 and Chapter 4. In Chapter 5, I introduce a mutual information maximization-based GNN for heterogeneous graph representation learning. Chapter 6 discusses my contributions to training larger and deeper GNNs through a subgraph-based learning framework. The other direction is the Application of GNNs in Real-world Topics. As an effective tool for processing graph data, GNNs being applied to solve real-world graph mining problems can further verify the effectiveness. Meanwhile, the application of GNNs requires a combination of domain knowledge and specific data modeling, which is also a challenge that needs to be addressed. In Chapter 7, I apply GNNs to the emerging and non-trivial topic of fake news detection. When dealing with the fake news detection topic, I innovate the GNNs model to handle the challenges of the fake news detection problem, which is critical for GNNs to exert the best effect. Experiments with real-world fake news data show that the novel GNN can outperform text-based models and other graph-based models, especially when using less labeled news data. In the last chapter, I provide concluding thoughts about this dissertation thesis.