Describing graphs

Here's one way to represent a social network:

A line between the names of two people means that they know each other. If there's no line between two names, then the people do not know each other. The relationship "know each other" goes both ways; for example, because Audrey knows Gayle, that means Gayle knows Audrey.

This social network is a graph. The names are the vertices of the graph. (If you're talking about just one of the vertices, it's a vertex.) Each line is an edge, connecting two vertices. We denote an edge connecting vertices $u$ and $v$ by the pair $(u, v)$. Because the "know each other" relationship goes both ways, this graph is undirected. An undirected edge $(u, v)$ ...