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Master the Nuts and Bolts of Graph Theory: the Heart of Communication and Transportation Networks, Internet, GPS, ...

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What Will I Learn?

- Master fundamental concepts in Graph Theory
- Get to know a wide range of different Graphs, and their properties.
- Be able to preform Elementary, Advanced Operations on Graphs to produce a new Graph
- Understand Graph Coloring.
- Understand Eulerian and Hamiltonian paths and circuits. And many related topics to Paths.
- Know how to turn a Graph into a Matrix and vice versa.
- Obtain a solid foundation in Trees, Tree Traversals, and Expression Trees.
- Have a good understanding of Graph Match.

Requirements

- To know elementary operations like addition and multiplication

Description

**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:

: In this section you will learn basic definitions like Vertex, Edge, Distance, Contentedness, and many other concepts that are the alphabet of Graph Theory.__Graphs__: In this section you will learn a variety of different Graphs, and their properties.__Types of Graphs__In this section you will learn different operations and different methods in making new Graphs.__Graph Operations:__in this section you will learn Graph Coloring and many related concepts.__Graph Coloring:__in this lecture you will learn Euler and Hamiltonian Paths and Circuits, and many other concepts in that area.__Paths:__**Trees:**In this section you will learn about Trees, Tree Traversals, Binary Expression Trees and some more.- And
**Graph Match:**In this section you will about Graph Match and Graph Cover.

**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 3-5 lectures so you can test what you have learned and go over something if needed.

Who is the target audience?

- Mathematics or Computer Science students
- Anyone interested in learning advanced Mathematics in an easy way

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Curriculum For This Course

Expand All 74 Lectures
Collapse All 74 Lectures
05:45:51

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Introduction
1 Lecture
02:04

Preview
02:04

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Graphs
12 Lectures
01:08:26

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.

Graphs

06:29

In this lecture we will get to know Subgraphs, and we will define Vertex Set and Edge Set.

Subgraphs

03:39

Quiz

4 questions

In this lecture we will explore Graph Isomorphism and its conditions.

Graph Isomorphism

03:25

Graph Automorphism

03:52

Quiz

3 questions

In this lecture we will talk about Complement Graphs, and what it means for Vertices to be adjacent.

Complement Graph

03:45

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

Multigraphs

01:48

Quiz

4 questions

In this lecture we will talk about Adjacency Matrix and Incidence Matrix.

Matrix Representation

13:41

Quiz

3 questions

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

Walks, Trails and Paths

08:08

In this lecture we will talk about how Distance is measured in Graphs.

Distance

05:49

In this lecture we will talk about Graph Connectedness, Cut Edge, Cut Vertex, and Separating Sets.

Connectedness

06:46

Quiz

6 questions

Menger's theorem

08:22

Sum of Degrees of Vertices Theorem

02:42

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Graph Types
11 Lectures
46:52

Preview
00:59

In this lecture we will define Null, Trivial, Simple Graphs, Loops, and Parallel Edges.

Null, Trivial, and Simple Graphs

03:49

Regular, Complete and Weighted Graphs

03:01

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.

Directed, Undirected and Mixed Graphs

07:52

Quiz

4 questions

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.

Cycle, Path, Wheel and Lolipop Graphs

08:13

Planar, Cubic and Random Graphs

04:05

In this lecture we will talk about Ladder and Prism Graphs, and how we can count the number of the Edges in each.

ladder and Prism Graphs

05:33

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

Web and Signed Graphs

05:43

Quiz

7 questions

Peterson Graph

00:55

Bipartite Graphs

03:31

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

Platonic Graphs

03:11

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Graph Operations
13 Lectures
50:31

Preview
02:19

Vertex Addition and Deletion

02:30

Edge Addition and Deletion

02:27

Vertex Contraction and Edge Contraction

05:06

Quiz

2 questions

Graph Minor and Graph Transpose

03:33

In this lecture we will talk about Line Graphs which, and the process of creating them.

Line Graphs

05:43

In this lecture we will talk about the process of creating a Dual Graph from another Graph.

Dual Graphs

04:57

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

Graph Power

03:26

Y - Δ Transform

02:26

Quiz

5 questions

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".

Graph Join and Graph Product

06:46

Hajós Construction

04:09

In this lecture we will talk about how we can create a new Graph by Union and Intersection of two Graphs.

Graph Union and Graph Intersection

04:18

Series - Parallel Composition

02:51

Quiz

4 questions

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Graph Coloring
9 Lectures
38:40

Preview
01:04

In this lecture we will define Vertex Coloring, Chromatic Number, k-Colorable Graphs, and Independent Sets.

Vertex Coloring

07:26

Edge Coloring

03:49

Quiz

3 questions

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

Chromatic Polynomial

05:31

Total and List Coloring

07:03

Exact and Fractional Coloring

03:59

Rainbow Coloring

03:11

In this lecture we will talk about Vizing's Theorem and Maximum Degree.

Vizing's Theorem

04:14

Four Color Theorem

02:23

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Paths
12 Lectures
01:04:12

Preview
01:17

In this lecture we will talk about what triggered Graph Theory.

The Königsberg Bridge Problem

01:43

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.

Euler Paths and Circuits

08:20

In this lecture we will talk about Fleury's way of finding an Euler Path or Circuit.

Fleury’s Algorithm

05:14

Hierholzer's Algorithm

10:48

Quiz

3 questions

In this lecture we will define Hamiltonian Paths and Circuits.

Hamiltonian Paths and Circuits

05:33

In this lecture we will explore decomposing a Graph based on the Hamiltonian Circuits in it.

Hamiltonian Decomposition

01:33

Ore's Theorem

04:28

Dirac's Theorem

03:30

Quiz

3 questions

In this lecture we will see how we can find the shortest path in a Graph using Dijkstra's Algorithm.

Shortest Path Problem

13:59

Five Room Puzzle

04:22

Knight's Tour

03:25

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Trees
10 Lectures
57:02

Preview
00:59

In this lecture we will define Trees, their properties, their importance in Computer Science, and how we can count them using Cayley's Formula.

Trees

05:26

In this lecture we will talk about Star Trees, Caterpillar Trees, Lobster Trees, and Banana Trees.

Tree Types

03:49

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.

Rooted Trees

06:24

In this lecture, we will talk about different ways we can represent a Tree visually.

Tree Structures

05:36

Quiz

5 questions

In this lecture we will talk about Binary Trees, and its different types (Proper, Perfect, Complete, Infinite Complete, and Balanced Binary Trees).

Binary Trees

07:50

Spanning Trees

04:44

Quiz

4 questions

In this lecture you will learn how to convert an Algebraic or Boolean expression into a Tree and vice versa.

Binary Expression Trees

06:36

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

Tree Traversal

14:02

Quiz

3 questions

Forests

01:36

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Graph Match
6 Lectures
18:04

Preview
00:32

In this lecture we will get to know Matching, Maximum and Maximal Matching, Perfect Matching, and Near Perfect Matching.

Graph Match

06:50

Preview
02:57

In this lecture, beside talking about Berge's Lemma, we will talk about Augmenting Paths, and Alternating Paths.

Berge's Lemma

02:30

Vertex and Edge Cover

03:53

In this lecture we will see the relationship between Matching and Vertex Cover for Bipartite Graphs.

König's Theorem

01:22

Quiz

5 questions

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Miran Fattah,
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B.S. in Mathematics & Geophysics

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About the Instructor

B.S. in Mathematics & Geophysics

Fattah has B.S. in **Mathematics **and** Geophysics** from the* University 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

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.

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