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

Incident
An edge $$e = \left\{{u, v}\right\} \in E$$ is said to be incident to $$u$$ and $$v$$.

By the same coin, if $$e = \left\{{u, v}\right\} \in E$$ then $$u$$ and $$v$$ are likewise incident to $$e$$.

Adjacent
Two vertices $$u$$ and $$v$$ are said to be adjacent or neighboring if there exists an edge $$e = \left\{{u, v}\right\} \in E$$ to which they are both incident.

Degree
The degree of a vertex is the number of edges to which it is incident.

That is:
 * $$\deg_G \left({v}\right) = \left|{\left\{{u \in V : \left\{{u, v}\right\} \in E}\right\}}\right|$$.

If the degree of a vertex $$v$$ is even, then $$v$$ is called an even vertex. If the degree of $$v$$ is odd, then $$v$$ is an odd vertex. If the degree of $$v$$ is zero, then $$v$$ is an isolated vertex.

The degree sequence of a graph is a list of the degrees of all the vertices of the graph in descending order.

Connected
A graph is connected if there is a path between any pair of vertices in the graph. Otherwise, it is called disconnected.

Note
Not to be confused with the Graph of a Mapping.