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What is this course about?
Graph Theory is an advanced topic in Mathematics. On a university level, this topic is taken by senior students majoring in Mathematics or Computer Science; however, this course will offer you the opportunity to obtain a solid foundation in Graph Theory in a very short period of time, AND without requiring you to have any advanced Mathematical background.
You don’t need to know complex Mathematical statements, or rules, but ALL you need to know is simple mathematical operations like addition and multiplication. The course is designed to be understood by an 11th grader since the structure of the course starts with the very basic idea of how to create a Graph, and with each step the ideas get more and more complex. The structure of the course goes as following starting with the first section:
How are the concepts delivered?
Each lecture is devoted to explaining a concept or multiples concepts related to the topic of that section. There are example(s) after the explanation(s) so you understand the material more. The course is taught in plain English, away from cloudy, complicated mathematical jargons and that is to help the student learn the material rather than getting stuck with fancy words.
How to learn better?
Take notes and repeat the lectures to comprehend the concepts more. Also, there are quizzes every 35 lectures so you can test what you have learned and go over something if needed.
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Lecture 1 
Introduction
Preview

02:04  
Section 1: Graphs  

Lecture 2  06:29  
In this lecture we will define Graphs, Vertices, Edges, Degree of a Vertex, Degree Sequence, Graph Order, and Graph Size. And we will get to know the importance of Graphs. 

Lecture 3  03:39  
In this lecture we will get to know Subgraphs, and we will define Vertex Set and Edge Set. 

Quiz 1 
Quiz

4 questions  
Lecture 4  03:25  
In this lecture we will explore Graph Isomorphism and its conditions. 

Lecture 5 
Graph Automorphism

03:52  
Quiz 2 
Quiz

3 questions  
Lecture 6  03:45  
In this lecture we will talk about Complement Graphs, and what it means for Vertices to be adjacent. 

Lecture 7  01:48  
In this lecture we will talk about what it means for two Vertices to be connected by more than one Edge. 

Quiz 3 
Quiz

4 questions  
Lecture 8  11:56  
In this lecture we will talk about Adjacency Matrix and Incidence Matrix. 

Quiz 4 
Quiz

3 questions  
Lecture 9  08:08  
In this lecture we will define Walks, Trails, and Paths, and the difference between them. Also, we will talk about Self Avoiding Paths (SAP). 

Lecture 10  05:49  
In this lecture we will talk about how Distance is measured in Graphs. 

Lecture 11  06:46  
In this lecture we will talk about Graph Connectedness, Cut Edge, Cut Vertex, and Separating Sets. 

Quiz 5 
Quiz

6 questions  
Lecture 12 
Menger's theorem

08:22  
Lecture 13 
Sum of Degrees of Vertices Theorem

03:38  
Section 2: Graph Types  
Lecture 14 
Introduction
Preview

00:59  
Lecture 15  03:49  
In this lecture we will define Null, Trivial, Simple Graphs, Loops, and Parallel Edges. 

Lecture 16 
Regular, Complete and Weighted Graphs

03:01  
Lecture 17  07:52  
In this lecture we will define Directed Graphs, Indegree, Outdgree, a Source, and a Sink, and we will learn how we can do Adjacency Matrix for a Directed Graph. We will also talk about Undirected, and Mixed Graphs. 

Quiz 6 
Quiz

4 questions  
Lecture 18  08:13  
In this lecture we will learn the difference between Cycle Graphs, and a Cycle in a Graph. We will also define Girth of a Graph, Path Graphs, Wheel Graphs, and Lollipop Graphs. 

Lecture 19 
Planar, Cubic and Random Graphs

04:05  
Lecture 20  05:33  
In this lecture we will talk about Ladder and Prism Graphs, and how we can count the number of the Edges in each. 

Lecture 21  05:43  
In this lecture we will define Web and Signed Graphs, and we will get to know a psychologist's contribution to Graph Theory. 

Quiz 7 
Quiz

7 questions  
Lecture 22 
Peterson Graph

00:55  
Lecture 23 
Bipartite Graphs

03:31  
Lecture 24  03:11  
The illustrations shown in this lecture are NOT owned by the instructor of this course. To reach the website containing the illustrations, follow this link : https://www.learner.org/interactives/geometry/platonic.html 

Section 3: Graph Operations  
Lecture 25 
Introduction
Preview

02:19  
Lecture 26 
Vertex Addition and Deletion

02:30  
Lecture 27 
Edge Addition and Deletion

02:27  
Lecture 28 
Vertex Contraction and Edge Contraction

05:06  
Quiz 8 
Quiz

2 questions  
Lecture 29 
Graph Minor and Graph Transpose

03:33  
Lecture 30  05:43  
In this lecture we will talk about Line Graphs which, and the process of creating them. 

Lecture 31  04:57  
In this lecture we will talk about the process of creating a Dual Graph from another Graph. 

Lecture 32  03:26  
In this lecture, we will talk about how to find the k^th Power of a Graph. 

Lecture 33 
Y  Δ Transform

02:26  
Quiz 9 
Quiz

5 questions  
Lecture 34  06:46  
In the lecture we will talk about the process of Joining and the steps that go into the Cartesian Product of two Graphs. In Cartesian Product of two graphs, I mention the word "multiply" which in this context means "Product" or "Cartesian Product". 

Lecture 35 
Hajós Construction

04:09  
Lecture 36  04:18  
In this lecture we will talk about how we can create a new Graph by Union and Intersection of two Graphs. 

Lecture 37 
Series  Parallel Composition

02:51  
Quiz 10 
Quiz

4 questions  
Section 4: Graph Coloring  
Lecture 38 
Introduction
Preview

01:04  
Lecture 39  07:26  
In this lecture we will define Vertex Coloring, Chromatic Number, kColorable Graphs, and Independent Sets. 

Lecture 40 
Edge Coloring

03:49  
Quiz 11 
Quiz

3 questions  
Lecture 41  05:31  
In this lecture we will define Chromatic Polynomials and show you how to use the software to find the Chromatic Polynomial of any Graph. Here is the link to Bob Weaver's website: http://www.mtholyoke.edu/~bweaver/vita/software.htm 

Lecture 42 
Total and List Coloring

07:03  
Lecture 43 
Exact and Fractional Coloring

03:59  
Lecture 44 
Rainbow Coloring

03:11  
Lecture 45  04:14  
In this lecture we will talk about Vizing's Theorem and Maximum Degree. 

Lecture 46 
Four Color Theorem

02:23  
Section 5: Paths  
Lecture 47 
Introduction
Preview

01:17  
Lecture 48  01:43  
In this lecture we will talk about what triggered Graph Theory. 

Lecture 49  08:20  
In this lecture we will define Euler Paths and Euler Circuits, and we will see why there isn't a solution to the Königsberg Bridge Problem. 

Lecture 50  05:14  
In this lecture we will talk about Fleury's way of finding an Euler Path or Circuit. 

Lecture 51 
Hierholzer's Algorithm

10:48  
Quiz 12 
Quiz

3 questions  
Lecture 52  05:33  
In this lecture we will define Hamiltonian Paths and Circuits. 

Lecture 53  01:33  
In this lecture we will explore decomposing a Graph based on the Hamiltonian Circuits in it. 

Lecture 54 
Ore's Theorem

04:28  
Lecture 55 
Dirac's Theorem

03:30  
Quiz 13 
Quiz

3 questions  
Lecture 56  13:59  
In this lecture we will see how we can find the shortest path in a Graph using Dijkstra's Algorithm. 

Lecture 57 
Five Room Puzzle

04:22  
Lecture 58 
Knight's Tour

03:25  
Section 6: Trees  
Lecture 59 
Introduction
Preview

00:59  
Lecture 60  05:26  
In this lecture we will define Trees, their properties, their importance in Computer Science, and how we can count them using Cayley's Formula. 

Lecture 61  03:49  
In this lecture we will talk about Star Trees, Caterpillar Trees, Lobster Trees, and Banana Trees. 

Lecture 62  06:24  
In this lecture we will define Rooted Trees, Out Tree, In Tree, Parent, Child, Sibling, Ancestor, Descendant, Uncle, Leaf, Internal and External Vertices, Subtree, Levels, Height, and Depth. 

Lecture 63  05:36  
In this lecture, we will talk about different ways we can represent a Tree visually. 

Quiz 14 
Quiz

5 questions  
Lecture 64  07:50  
In this lecture we will talk about Binary Trees, and its different types (Proper, Perfect, Complete, Infinite Complete, and Balanced Binary Trees). 

Lecture 65 
Spanning Trees

04:44  
Quiz 15 
Quiz

4 questions  
Lecture 66  06:36  
In this lecture you will learn how to convert an Algebraic or Boolean expression into a Tree and vice versa. 

Lecture 67  14:02  
In this lecture we will talk about Preorder, Inorder, Postorder, and Levelorder Tree Traversals. To practice more, go to below website and you will find numerous practice examples at the very end of the page. http://algoviz.org/OpenDSA/Books/OpenDSA/html/BinaryTreeTraversal.html 

Quiz 16 
Quiz

3 questions  
Lecture 68 
Forests

01:36  
Section 7: Graph Match  
Lecture 69 
Introduction
Preview

00:32  
Lecture 70  06:50  
In this lecture we will get to know Matching, Maximum and Maximal Matching, Perfect Matching, and Near Perfect Matching. 

Lecture 71 
Hosoya Index
Preview

02:57  
Lecture 72  02:30  
In this lecture, beside talking about Berge's Lemma, we will talk about Augmenting Paths, and Alternating Paths. 

Lecture 73 
Vertex and Edge Cover

03:53  
Lecture 74  01:22  
In this lecture we will see the relationship between Matching and Vertex Cover for Bipartite Graphs. 

Quiz 17 
Quiz

5 questions 
Fattah has B.S. in Mathematics and Geophysics from theUniversity of Oklahoma in Oklahoma, USA. He has taught and tutored many college students both in the United States and Iraq. His love for teaching made him one of four students in Iraq to receive a full scholarship to pursue a B.S. degree in the States so to return back to his home country and teach.
He is passionate about Math & Science and loves to share his passion with others. To him, Mathematics and Sciences are crucial for everyone to learn no matter how little. He is a BIG believer in visual learning, and his aim is to deliver the concepts in an easy and direct way so as to make the learning process fast for everyone.