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Graph theory is a branch of mathematics that studies relationships and structures known as graphs, which are made up of nodes (also referred to as vertices) and edges. It provides a way to represent and analyze various problems in a visual and conceptual format, from social networks to computer networks, transport systems, and more. In our case with John, Joan, Jean, Jane, and Joe, graph theory enables us to visualize the social interactions between these individuals effectively.

Understanding graph theory starts with recognizing its fundamental elements: vertices or nodes, which represent points in the graph, and edges, the connections between these points. In the provided exercise, individuals are represented as vertices, while their affections, or 'likes', serve as the connections or edges between them. By applying graph theory, we can create a diagram, or graph, that provides insight into the complexity of these relationships in a clear and manageable way.

A directed graph, or digraph, is a type of graph where edges have a direction, indicating a one-way relationship between two vertices. In contrast to an undirected graph where edges imply a two-way, mutual connection, a directed graph has arrows that show which vertex has an influence or relationship with the other.

In our scenario, a directed graph is perfect for illustrating one-way affections such as 'John likes Joan'. The arrow points from John to Joan, clearly indicating the direction of affection. Digraphs are instrumental in representing systems where flow or directionality is essential, such as in computer science for representing dependencies or in sociograms to represent social dynamics. They can also depict hierarchies and processes, providing a tool for decision-making and analysis in various fields.

Vertices, or nodes, are the cornerstone of any graph in graph theory. They act as fundamental units through which the relationships are established by edges. Each vertex typically represents an entity or a point within the graph's domain of discourse.

In the context of our exercise, the vertices represent the individuals: John, Joan, Jean, Jane, and Joe. When illustrating relationships in a social network as a directed graph, each 'like' adds a directed edge between the corresponding vertices. For instance, if John likes Joan, this is reflected by an arrow going from the vertex representing John to the one representing Joan. Even when mutual liking occurs, like between Jean and Joan, two directed edges are drawn, one from each vertex to the other, to reflect the directionality of each individual's feelings. As we navigate through graph theory, the role of vertices is pivotal as they provide the foundational structure from which the graph's shape and meaning are derived.