Definition:Graph (Graph Theory)

Definition
A graph is intuitively defined as a pair consisting of a set of 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
In the above, the edges are $AB, AE, BE, CD, CE, CF, DE, DF, FG$.

Also see

 * Definition:Simple Graph: a graph which:
 * has no more than one edge between a given pair of vertices
 * in which the edges start and end on different vertices.


 * Definition:Multigraph: A graph which may have more than one edge between a given pair of vertices


 * Definition:Loop-Graph: A graph which allows an edge to start and end at the same vertex
 * Definition:Loop (Graph Theory)


 * Definition:Loop-Multigraph: A multigraph which is allowed to have loops


 * Definition:Directed Graph or Definition:Digraph: A graph in which the edges are ordered pairs of vertices.


 * Definition:Weighted Graph, where in addition the edges are each mapped to a number


 * Definition:Null Graph: A graph whose vertex set is empty.

Also defined as
Many treatments of this subject require that $V$ is non-empty, and so do not recognise the concept of a null graph.