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Graph theory is a field of mathematics that deals with the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph is made up of vertices (also called nodes or points) and edges (lines or links) that connect some pairs of vertices.

In more practical terms, graph theory applies to various systems in the real world such as social networks, where individuals are represented as vertices and their relationships as edges, or transportation networks, where locations are vertices and roads or pathways are edges. Graph theory is foundational to computer science, as it can represent computer networks, data organization, and computational problems.

To understand the various types of graphs, one must grasp some basic concepts and terminology such as 'complete graphs', 'subgraphs', 'directed graphs', and many others, which denote specific properties and structures within a graph.

Edges are one of the two fundamental units in graph theory that represent the relationships or connections between vertices in a graph. If you consider a graph as a network of interconnected points, edges are the lines that link these points together.

An edge can be either directed or undirected. Directed edges, also known as arcs or arrows, have a direction, showing a one-way relationship, while undirected edges show a two-way, mutual relationship. The number of edges incident on a vertex defines its degree, and this is a critical concept in understanding the structure of graphs. For example, in a complete graph, each vertex has an edge to every other vertex, making the degree of each vertex directly related to the total number of vertices.

Understanding how edges function in a graph helps to analyze and solve problems related to connectivity, traversability, and network flow.

Vertices, also known as nodes or points, are the fundamental units in graph theory that represent entities or locations within a graph structure. Each vertex in a graph is a distinct entity, and it can be connected to other vertices via edges.

The number of edges converging at a vertex is known as its 'degree'. For example, in a complete graph where every vertex is connected to every other vertex, the degree of each vertex is one less than the total number of vertices in the graph \( n-1 \). Vertices can have attributes like names or values, adding more information to the graph representation.

In real-world applications, vertices can represent computers in a network, intersections in a road map, or individual people in a social network analysis. The study of vertices and their connections is crucial to interpreting graphs and understanding the underlying systems they model.