Notation

Graph

We define a simple, unlabelled graph G as a pair of two sets:

$$G\stackrel{\text{def}}{=}(N,E)$$

Where N is a set of nodes and E is a set of edges between the nodes in the graph. $N$ can also be denoted $V$, which denotes vertex. They are interchangable.

Nodes

N can be defined by a set of pronumerals.

$$N=\{A,B,C...\}$$

or

$$N=\{n_1,n_2,n_3....\}$$

where:

$$|N|=\text{ Total number of nodes in graph}$$

Edges

E can be defined as a set of unordered pairs $\{n_1,n_2\}$ where $n_1$ and $n_2$ are called the endpoints of the edge:

$$E\subseteq \{\{n_1,n_2\}:n_1,n_2\in N\}$$

where:

$$|E|=\text{ Total number of edges in graph}$$

Degree

The degree of a node with 1-based index $i$ in the set of all nodes $N$ is denoted:

$$Deg(N[i])=\text{The degree of the i-th node in the set of all nodes }$$