uni/year2/semester1/logseq-stuff/pages/Introduction to Graph Theory.md

• #MA284 - Discrete Mathematics
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• Introduction to Graph Theory

  • What is a graph? #card card-last-interval:: 4 card-repeats:: 2 card-ease-factor:: 2.7 card-next-schedule:: 2022-11-18T20:20:08.384Z card-last-reviewed:: 2022-11-14T20:20:08.384Z card-last-score:: 5
    • 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 card-last-interval:: 4 card-repeats:: 2 card-ease-factor:: 2.7 card-next-schedule:: 2022-11-18T20:19:56.959Z card-last-reviewed:: 2022-11-14T20:19:56.959Z card-last-score:: 5
    • 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 ```

• Graph Theory - The Basics

  • What is the order of a graph? #card card-last-interval:: 4 card-repeats:: 2 card-ease-factor:: 2.7 card-next-schedule:: 2022-11-18T20:20:23.073Z card-last-reviewed:: 2022-11-14T20:20:23.074Z card-last-score:: 5
    • The order of a graph G = (V,E) is the size of its vertex set, |V|.

  • Equality & Isomorphism

    • What makes two graphs equal? #card card-last-interval:: 4 card-repeats:: 2 card-ease-factor:: 2.7 card-next-schedule:: 2022-11-18T20:20:18.088Z card-last-reviewed:: 2022-11-14T20:20:18.088Z card-last-score:: 5
      • 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 card-last-interval:: 4 card-repeats:: 2 card-ease-factor:: 2.22 card-next-schedule:: 2022-11-18T20:17:59.530Z card-last-reviewed:: 2022-11-14T20:17:59.530Z card-last-score:: 3
      • 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.
  • What is a simple graph? #card card-last-interval:: 2.97 card-repeats:: 2 card-ease-factor:: 2.6 card-next-schedule:: 2022-11-17T19:16:42.421Z card-last-reviewed:: 2022-11-14T20:16:42.421Z card-last-score:: 5
    • 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 card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T20:14:08.619Z card-last-score:: 1
    • 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.