Lecture Notes on 12 Apr 2023

* Euler Characteristic Equation
  V - E + F = 2

* Five Regular Solids
  - Tetrahedron: 4 (V), 6 (E), 4 (F)
  - Cube: 8 (V), 12 (E), 6 (F)
  - Octahedron: 6 (V), 12 (E), 8 (F)
  - Dodecahedron: 20 (V), 30 (E), 12 (F)
  - Icosahedron: 12 (V), 30 (E), 20 (F)

  - Download images of the Five Regular Solids
https://www.cs.utexas.edu/users/mitra/csSpring2023/cs313/notes/platonic_solids.jpeg

* Graph Implementation
  - Download the map of cities in the US
https://www.cs.utexas.edu/users/mitra/csSpring2023/cs313/notes/US_Cities.pdf
  - Download the textual representation of this graph
https://www.cs.utexas.edu/users/mitra/csSpring2023/cs313/notes/graph.txt

* Minimum Spanning Tree (MST) - Kruskal's Algorithm and Prim's
  Algorithm.

* Kruskal's Algorithm
  - Order the edges by increasing weight.
  - Start with an empty graph having only the vertices but no
    edges. Add an edge to this graph as long as it does not form 
    a cycle.
  - When all the vertices are connected you are done.

* Prim's Algorithm
  - Start with an empty graph having only the vertices.
  - Start with any vertex and add it to the list of visited vertices.
  - Choose the edge with the smallest weight to an unvisited vertex
    as long as it does not form a cycle. Add that vertex to the list
    of visited vertices.
  - Keep adding the edges with the smallest weight from any of the
    vertices in the list of visited vertices as long as those edges
    do not form a cycle.
  - When all the vertices have been visited then you are done.

* Do Exercises on Graphs

https://www.cs.utexas.edu/users/mitra/csSpring2023/cs313/notes/Graph_Exercises.pdf

https://www.cs.utexas.edu/users/mitra/csSpring2023/cs313/notes/Minimum_Spanning_Tree.pdf