1.0 - Graphs

Graphs are comprised of vertices and edges.

Vertices $\{A, B, C, D, E, F\}$
Edges $\{(A, B), (A, D), (A, C), (B, D), (C, D), \\(E, D), (E, F), (D, F) \}$

Could be used to represent different types of information

Geographical information (Vertices = cities, Edges = roads)
Networks (Vertices = computers, Edges = cables)
Brains (Vertices = neurons, Edges = synapses)
Programs (Vertices = statements, Edges = control flow)

1.1 - Directed or Undirected Graphs

1.1.1 - Undirected Graphs

Cannot have self-loops
If $(u, v)\in E(G)$ , $v$ is adjacent to u in G (and u is adjacent to v) - symmetric property
Edge $(u,v)\in E(G)$ is incident on vertices $v, u$

- The `degree` of a vertex $v$ is the number of edges incident upon it. - Maximum number of edges $\sum^{n-1}_{i=1}i=\frac{n(n-1)}{2}$ (as every vertex can have an edge to every other edge).

1.1.2 - Directed Graphs

Can have self-loops
If $(u,v )\in E(G)$ , $v$ is adjacent to $u$ - asymmetric property
Edge $(u, v)\in E(G)$ is incident from (leaves) vertex $u$ and incident on (enters) vertex $v$

The out-degree of a vertex $v$ is the number of edges leaving it
The in-degree of a vertex v is the number of edges entering it
The degree of a vertex is the sum of its in-degree and out-degree.
Maximum number of edges $n^2$

1.2 - Weighted or Unweighted Graphs

1.2.1 - Unweighted Graphs

A graph in which edges don’t have weights associated with them

1.2.2 - Weighted Graph

A graph in which each edge has a weight associated with it.

1.3 - Paths

For a graph $G=(V, E)$

A path of length $k$ $k$ from a vertex $v_0$ $v_{0}$ to a vertex $v_k$ $v_{k}$ is a sequence $\langle v_0, v_1, \cdots, v_k\rangle$ $⟨ v_{0}, v_{1}, \dots, v_{k} ⟩$ of vertices such that $(v_{i-1}, v_i)\in E$ $(v_{i - 1}, v_{i}) \in E$ for $i\in\{1, 2, \cdots, k\}$ $i \in {1, 2, \dots, k}$
- That is, every two consecutive vertices form an edge that is in the graph.
$u$ is reachable from $v$ if there is path from $v$ to $u$
A path is simple if all of the vertices i nthe path are distinct
A path forms a cycle if $v_0=v_k$ and $k>1$
A cycle is simple if $v_1, \cdots, v_k$ are distinct, and all of its edges are distinct
A graph with no simple cycles is acyclic

1.4 - Undirected Graphs - More Properties

An undirected graph is connected if every vertex is reachable from all vertices
- An undirected graph is connected if every vertex is reachable from all other vertices
- An undirected graph is a forest if it is acyclic
- An undirected graph is a tree if it is a forest (i.e. acyclic) with only one connected component.

What are the minimum and maximum number of edges in a:

Tree with $n$ vertices? $(n-1, n-1)$
Forest with $n$ $n$ vertices? $(0, n-1)$ $(0, n - 1)$
- Doesn’t have to be connected, so minimum is 0
Connected graph with $n$ vertices? $(n-1, \frac{n(n-1)}{2})$

1.5 - Directed Graphs - More Properties

A directed graph $G=(V, E)$ is:

Strongly connected if every two vertices are reachable from each other

1.6 - Graph Terminology

Graph $G'=(V', E')$ is a sub-graph of $G=(V, E)$ when $V'\subseteq V$ and $E'\subseteq E$
$G'$ is a spanning sub-graph of $G$ if the above, and $V'=V$ also holds.
The sub-graph of $G=(V, E)$ that is induced by $V'$ is the graph $G'=(V', E')$ where:

$E'=\{(u, v)\in E\ :\ u \in V'\wedge v \in V'\}$