**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.

### Graphs

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.

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

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

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

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

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

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

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

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

### Graph Types

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

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.

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.

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

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

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

### Graph Operations

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

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

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

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".

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

### Graph Coloring

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

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

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

### Paths

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

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.

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

In this lecture we will define Hamiltonian Paths and Circuits.

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

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

### Trees

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

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

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.

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

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

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

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

### Graph Match

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

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

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