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WHAT IS A GRAPH?
0:00:00 Airlines Graph
0:01:27 Knight Transposition
0:03:42 Seven Bridges of Königsberg
0:08:04 What is a Graph
0:15:23 Graph Example
0:17:52 Graph Applications
0:21:17 Vertex Degree
0:24:54 Paths
0:30:13 Connectivity
0:33:01 Directed Graphs
0:36:17 Weighted Graphs
0:38:31 Paths,Cycles and Complete Graphs
0:41:27 Trees
0:48:14 Bipartite Graphs

CYCLES
0:52:33 Handshaking Lemma
0:59:55 Total Degree
1:04:57 Connected Components
1:12:16 Guarini PUzzle Code
1:19:00 Lower Bound
1:24:32 The Heaviest Stone
1:31:14 Directed Acyclic Graphs
1:41:23 Strongly Connected Components
1:48:56 Eulerian Cycles
1:53:05 Eulerian Cycles Criteria
2:04:50 Hamitonian Cycles
2:09:08 Genome Assembly

GRAPH CLASSES
2:22:00 Road Repair
2:25:47 Trees
2:33:49 Minimum Spanning Tree
2:40:19 Job Assigment
2:44:07 Biparitite Graphs
2:49:28 Matchings
2:53:24 Hall's Theorem
3:00:59 Subway Lines
3:02:21 Planar Graphs
3:05:34 Eular's Formula
3:09:53 Applications of Euler's Formula

GRAPH PARAMETERS
3:17:01 Map Coloring
3:20:36 Graph Coloring
3:23:42 Bounds on the Chromatic Number
3:27:36 Applications
3:30:45 Graph Cliques
3:34:17 Clique and Independent Sets
3:37:31 Connections to Coloring
3:39:05 Mantel's Theorem
3:44:17 Balanced Graphs
3:46:47 Ramsey Numbers
3:49:04 Existence of Ramsey Numbers
3:55:01 Antivirus System
3:57:03 Vertex Covers
4:00:39 König's Theorem

FLOW AND MATCHINGS
4:08:59 An Example
4:15:42 The Framwork
4:24:00 Ford and Fulkerson Proof
4:35:36 Hall's Theorem
4:45:43 What Else
4:54:14 Why Stable Matchings
5:00:20 Mathematics and REal life
5:04:53 Basic Examples
5:11:36 Looking for a Stable Matching
5:17:51 Gale-Shapley Algorithm
5:24:41 Correctness Proof
5:30:48 why The Algorithm is Unfair
5:38:47 why the Algorithm is Very unfair

We invite you to a fascinating journey into Graph Theory — an area which connects the elegance of painting and the rigor of mathematics; is simple, but not unsophisticated. Graph Theory gives us, both an easy way to pictorially represent many major mathematical results, and insights into the deep theories behind them.

In this course, among other intriguing applications, we will see how GPS systems find shortest routes, how engineers design integrated circuits, how biologists assemble genomes, why a political map can always be colored using a few colors. We will study Ramsey Theory which proves that in a large system, complete disorder is impossible!

By the end of the course, we will implement an algorithm which finds an optimal assignment of students to schools. This algorithm, developed by David Gale and Lloyd S. Shapley, was later recognized by the conferral of Nobel Prize in Economics.

As prerequisites we assume only basic math (e.g., we expect you to know what is a square or how to add fractions), basic programming in python (functions, loops, recursion), common sense and curiosity. Our intended audience are all people that work or plan to work in IT, starting from motivated high school students.

⭐ Important Notes ⭐
⌨️ This course is created in collaboration with University of California SAN DIEGO

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