The information age has led to an explosion in the size and availability of data. This data often exhibits graph-structure that is either explicitly defined, as in the web of a social network, or is implicitly defined and can be determined by measuring similarity between objects. Utilizing this graph-structure allows for the design of machine learning algorithms that reflect not only the attributes of individual objects but their relationships to every other object in the domain as well. This thesis investigates three machine learning problems and proposes novel methods that leverage the graph-structure inherent in the tasks. Quantum walk neural networks are classical neural nets that use quantum random walks for classifying and regressing on graphs. Asymmetric directed node embeddings are another neural network architecture designed to embed the nodes of a directed graph into a vector space. Filtered manifold alignment is a novel two-step approach to domain adaptation.