$$ E(G)=\{\langle v1, v2\rangle\, \langle v1, v4\rangle\,...,\langle v3, v6\rangle\} $$
$$ V(G)=\{v_1, v_2, ..., v_6\} $$
Neighbor set of $v\in V(G)$ is set of all vertices, except for $v$, that are adjacent to $v$.
$$ N(v)=\{u\in V(G)-\{u\}|\langle v, u\rangle \in E(G)\} $$
Example
$N(v_4)=\{v_1, v_2, v_5\}$
A graph is called a simple graph if it ⇒ does not contain multiple edges between the same pair of vertices ⇒ does not contain loops
Degree of vertex $v\in V(G)$ is the number of edges in $G$ incident (connected to) $v$. ⇒ Denoted by $\delta(v)$ or $\deg(v)$ Note - Loops are counted twice
In any graph $G$ we have $\displaystyle\Sigma_{v\in V(G)}\delta(v)=2|E(G)|$
Given an ordered degree sequence $D$ we know $\forall \deg(v)\in D$ that $0\leq \deg(v)\leq n-2$ OR $1\leq \deg(v)\leq n-1$ ⇒ Must be at lest 2 vertexes in a simple graph with the same degree
The degree sequence of a graph G is a list of the degrees of the vertices of G. The sequence is ordered if the numbers are in non-decreasing order. (start smallest end largest)
Example
$$ [2, 3, 3, 4] $$
A sequence of numbers is called graphic if it is the degree sequence of a simple graph.
Min degree of $G$ is the smallest degree in the graph $G$
$\Delta(G)$ is the largest degree that exists in $G$