Edge of a graph
Elementary
Imagine a network drawn as dots connected by lines, like a map of cities and roads. Each line connecting two dots in the network is called an edge.
Undergraduate (graph theory)
Consider a graph, made of a set of vertices and a set of edges. An edge of the graph is an element of its edge set. In a simple undirected graph, an edge is an unordered pair $\setdef{u, v}$ of distinct vertices.
Examples
Non-examples
Things that may be mistaken for examples.
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The unit circle, for the closed unit disk in $\reals^2$
High school
The unit circle is not an edge of a graph, because it is not a link between two vertices.
Notes
A graph edge has no position or length. Drawing it as a line segment is only a convenience.