🏗️ $\Theta \rho ϵ\eta \Pi \alpha \tau \pi$🚧 (under construction)

A graph $G$ is an ordered pair $V,E$ of sets $V$ of vertices and $E$ of edges, that are disjoint together with an incedence function $\varphi :E\to V\times V$ that associates with

Suppose that $G=(V,E)$ is a graph, then we say that the degree of a vertex $v$ is the number of other vertices connected to it with an edge. We define the syntax $deg:V\to {\mathbb{N}}^0$ to be the function that maps a node to it's degree.

Suppose that we have a graph $G$, then we say that $G$ has branching factor $b\in {\mathbb{N}}_1$, if every vertex has degree $b$

Suppose that $G$ is a finite graph, then the average branching factor of the graph is defined as $${\displaystyle \frac{{\sum }_{v\in V}deg\left(v\right)}{\left|V\right|}\in \mathbb{Q}}$$

Suppose that we have a graph $G$, then we say that $G$ has a maximum branching factor of $b\in {\mathbb{N}}_0$, if every vertex has degree at most $b$

A tuple $(v_1,v_2,\dots v_n)\in V^n$ is said to be a path if for every $i\in [1\dots n-1]$, $\{v_i,v_{i+1}\}\in E$.

Suppose that $u,v\in V$, then we say that $u,v$ are connected if there is a path $v_1,v_2,\dots ,v_n$ where $v_1=u$ and $v_n=v$

Given a graph $G$ we say that it is connected when for every $u,v\in V$, we have that $u$ and $v$ are connected.