Edges

Edges are a fundamental component in several disciplines, most prominently in graph theory and geometry. In graph theory, an edge is a connection between two vertices, representing a relationship or interaction. In an undirected graph, the edge has no direction; in a directed graph or arc, the edge has an origin and a destination. Edges may be unweighted, carrying no quantitative value, or weighted, assigning a numerical cost or length to the connection.

In graph terminology, several special cases exist. A simple graph has at most one edge between any pair of vertices and contains no loops. A multigraph allows multiple parallel edges between the same pair of vertices, while a pseudograph allows loops that connect a vertex to itself. The size of a graph is the number of edges, while the degree of a vertex counts how many incident edges it has (loops contributing twice to the degree in many definitions).

In geometry, the term edge refers to a line segment where two faces meet, such as the edges of a polyhedron or the sides of a polygon. In three-dimensional objects, edges meet at vertices and characterize the object's shape and size. The term can also describe the boundary between two regions in a mesh or surface in computer graphics and geometric modeling.

Edges underpin many algorithms and models, including shortest-path, network flow, and minimum spanning tree computations, as well as mesh generation and 3D rendering. They provide a compact abstraction for relationships in social networks, transportation, biology, and information systems, where a second value such as weight or capacity often informs optimization or analysis.