Definition:Graph (Graph Theory)

Informal Definition
A graph is intuitively defined as a pair consisting of a set of nodes or vertices and a set of edges.



Vertex
In the above, the vertices (singular: vertex) are the points $$A, B, C, D, E, F, G$$ which are marked as dots.

Edge
The edges are the lines that join the vertices together.

In the above, the edges are $$AB, AE, BE, CD, CE, CF, DE, DF, FG$$.

Formal Definition
More precisely, a graph is an ordered pair $$G = \left({V, E}\right)$$ such that:
 * $$V$$ is a set;
 * $$E$$ is a set of 2-element subsets of $$V$$.

That is: $$E \subseteq \left\{{\left\{{u, v}\right\}: u, v \in V}\right\}$$

An edge $$e = \left\{{u, v}\right\} \in E$$ is said to be incident to $$u$$ and $$v$$; similarly, $$u$$ and $$v$$ are said to be adjacent or neighboring.