- #[[MA284 - Discrete Mathematics]]
- **Previous Topic:** [[Advanced PIE, Derangements, & Counting Functions]]
- **Next Topic:** [[Definitions & Planar Graphs]]
- **Relevant Slides:** 
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- # Introduction to Graph Theory
- What is a **graph**? #card
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- A **graph** is a collection of:
- **vertices** (or "nodes"), which are the "dots" in the belong diagram.
- **edges** joining a pair of vertices.
- 
- If the graph is called $G$, we often define it in terms of its **edge set** $E$, and **vertex set** $V$ as
- $$G = (V,E)$$
- What are **adjacent** vertices? #card
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- If two vertices are connected by an edge, we say that they are **adjacent**.
- ## Example
- **Aoife, Brian, Conor, David, * Edel are students in an *Indescrete Mathematics* module.**
- **Aoife & Conor worked together on the assignment.**
- **Brian & David also worked together on their assignment.**
- **Edel helped everyone with their assignments.**
- **Represent this situation with a graph.**
- Let the students be vertices $A$, $B$, $C$, $D$, $E$. An edge represents collaboration between students.
- $$V = \{A,B,C,D,E\}$$
- $$ E = \{\{A,C\}, \{B,D\}, \{E,A\}, \{E,B\}, \{E,C\}, \{E,D\}\}$$
- 
{{renderer :mermaid_fzjdzhfgs}}
- ```mermaid
flowchart RL
A((A)) --- C((C))
B((B)) --- D((D))
E((E)) --- A
E --- B
E --- C
E --- D
```
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- # Graph Theory - The Basics
- What is the **order** of a graph? #card
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- The **order** of a graph $G = (V,E)$ is the size of its vertex set, $|V|$.
- ## Equality & Isomorphism
- What makes two graphs **equal**? #card
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- Two graphs are **equal** if they have exactly the same Edge & Vertex sets.
- That is, ^^it is not important how we draw them^^.
- What is an **Isomorphism**? #card
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- An **isomorphism** between two graphs, $G_1 = (V_1, E_1)$ & $G_2 = (V_2, E_2)$, is a **bijection** $f: V_1 \rightarrow V_2$ between the vertices in the graph such that, if $\{a, b\}$ is an edge in $G_1$, then $\{f(a), f(b)\}$ is an edge in $G_2$
- Two graphs are **isomorphic** if there is an isomorphism between them.
- In that case, we write:
- $$G_1 \cong G_2$$
- ### Example
- **Show that the graphs**
- $$G_1 = \{V_1, E_1\}, \text{ where } V_1 = \{a,b,c\} \text{, and } E_1 = \{\{a,n\}, \{a,c\}, \{b,c\}\};$$
- $$G_2 = \{V_2, E_2\}, \text{ where } V_2 = \{u,v,w\} \text{, and } E_2 = \{\{u,v\}, \{u,w\}, \{v,w\}\};$$
- **are not *equal* but are *isomorphic*. **
- $V_1 \neq V_2$ so graphs are not equal.
- 
{{renderer :mermaid_sewggjymkp}}
- ```mermaid
flowchart TB
subgraph G2
u((u)) --- v((v)) --- w((w))
end
subgraph G1
a((a)) --- b((b)) --- c((c))
end
```
- $f: V_1 \rightarrow V_2$ given by $f(a) = u$, $f(b) = v$, $f(c) = w$ is an isomorphism.
- e.g., $\{a,c\} \in E_1$, and $\{f(a), f(c)\} = \{u, w\} \in E_2$.
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- What is a **simple graph**? #card
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- A **simple graph** is one that:
- 1. has no **loops** (i.e., no edge from a vertex to itself).
2. have no repeated edges (i.e., there is at most one edge between each pair of vertices).
- Because simple graphs are so common, usually when we say "graph" we mean "simple graph", unless otherwise stated.
- What is a **multigraph**? #card
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- A **multigraph** is a graph that does have repeated edges.
- In a **multirgraph**, the list of edges is not a set, as some elements are repeated. It is a **multiset**.
-