Chapter 1: Introduction to Graph Theory
- Lesson 1: What is Graph Theory?
- Lesson 2: Historical Background and Applications
- Lesson 3: Graphs in Everyday Life
- Lesson 4: Basic Terminology (Vertices, Edges, Degree, etc.)
- Lesson 5: Overview of the Course
Chapter 2: Types of Graphs
- Lesson 1: Simple Graphs and Multigraphs
- Lesson 2: Directed and Undirected Graphs
- Lesson 3: Weighted and Unweighted Graphs
- Lesson 4: Complete, Bipartite, and Regular Graphs
- Lesson 5: Real-World Examples of Graph Types
Chapter 3: Representations of Graphs
- Lesson 1: Adjacency Matrix
- Lesson 2: Incidence Matrix
- Lesson 3: Adjacency List
- Lesson 4: Graph Visualization Tools
- Lesson 5: Comparing Representations
Chapter 4: Basic Graph Properties
- Lesson 1: Degree of a Vertex
- Lesson 2: Paths, Cycles, and Connectivity
- Lesson 3: Components of a Graph
- Lesson 4: Trees and Forests
- Lesson 5: Graph Density
Chapter 5: Eulerian and Hamiltonian Graphs
- Lesson 1: Eulerian Circuits and Trails
- Lesson 2: Conditions for Eulerian Graphs
- Lesson 3: Hamiltonian Paths and Cycles
- Lesson 4: Applications of Eulerian and Hamiltonian Graphs
- Lesson 5: Solving Problems
Chapter 6: Graph Traversal Algorithms
- Lesson 1: Depth-First Search (DFS)
- Lesson 2: Breadth-First Search (BFS)
- Lesson 3: Comparison of DFS and BFS
- Lesson 4: Applications in Problem Solving
- Lesson 5: Complexity of Traversal Algorithms
Chapter 7: Trees in Graph Theory
- Lesson 1: Properties of Trees
- Lesson 2: Rooted Trees and Binary Trees
- Lesson 3: Spanning Trees and Minimum Spanning Trees
- Lesson 4: Prim’s Algorithm for MST
- Lesson 5: Kruskal’s Algorithm for MST
Chapter 8: Graph Coloring
- Lesson 1: Introduction to Graph Coloring
- Lesson 2: Chromatic Number and Chromatic Polynomial
- Lesson 3: Applications of Graph Coloring (Scheduling, Map Coloring)
- Lesson 4: Greedy Coloring Algorithm
- Lesson 5: Limitations and Challenges
Chapter 9: Planar Graphs
- Lesson 1: Introduction to Planarity
- Lesson 2: Euler’s Formula for Planar Graphs
- Lesson 3: Kuratowski’s Theorem
- Lesson 4: Applications of Planar Graphs
- Lesson 5: Planarity Testing
Chapter 10: Matching in Graphs
- Lesson 1: What is a Matching?
- Lesson 2: Maximum Matchings and Perfect Matchings
- Lesson 3: Hall’s Marriage Theorem
- Lesson 4: Applications of Matchings
- Lesson 5: Algorithms for Matchings
Chapter 11: Connectivity and Network Analysis
- Lesson 1: Connected and Disconnected Graphs
- Lesson 2: Bridges and Articulation Points
- Lesson 3: Strong and Weak Connectivity
- Lesson 4: Applications in Network Design
- Lesson 5: Case Studies
Chapter 12: Introduction to Network Flows
- Lesson 1: Understanding Network Flow Problems
- Lesson 2: Max-Flow Min-Cut Theorem
- Lesson 3: Ford-Fulkerson Algorithm
- Lesson 4: Applications in Real-World Networks
- Lesson 5: Solving Flow Problems
Chapter 13: Directed Acyclic Graphs (DAGs)
- Lesson 1: What is a DAG?
- Lesson 2: Topological Sorting
- Lesson 3: Applications of DAGs in Scheduling
- Lesson 4: Longest Path Problem in DAGs
- Lesson 5: Real-World Examples
Chapter 14: Shortest Path Algorithms
- Lesson 1: Dijkstra’s Algorithm
- Lesson 2: Bellman-Ford Algorithm
- Lesson 3: Comparison of Algorithms
- Lesson 4: Applications in Transportation Networks
- Lesson 5: Problem Solving with Shortest Paths
Chapter 15: Bipartite Graphs
- Lesson 1: Properties of Bipartite Graphs
- Lesson 2: Applications in Matching Problems
- Lesson 3: Testing for Bipartite Graphs
- Lesson 4: Graph Coloring in Bipartite Graphs
- Lesson 5: Solving Real-Life Problems
Chapter 16: Line Graphs and Subgraphs
- Lesson 1: What is a Line Graph?
- Lesson 2: Properties and Applications
- Lesson 3: Subgraphs and Their Types
- Lesson 4: Induced Subgraphs
- Lesson 5: Graph Decomposition
Chapter 17: Graph Isomorphism
- Lesson 1: What is Graph Isomorphism?
- Lesson 2: Testing for Isomorphism
- Lesson 3: Applications of Isomorphism Testing
- Lesson 4: Complexity of Graph Isomorphism Problems
- Lesson 5: Practical Use Cases
Chapter 18: Applications of Graph Theory
- Lesson 1: Graph Theory in Computer Science
- Lesson 2: Social Network Analysis
- Lesson 3: Transportation Networks
- Lesson 4: Telecommunications Networks
- Lesson 5: Case Studies and Examples
Chapter 19: Introduction to Graph Algorithms in Programming
- Lesson 1: Basics of Implementing Graphs in Code
- Lesson 2: Graph Traversal in Python (or other languages)
- Lesson 3: Implementing Pathfinding Algorithms
- Lesson 4: Implementing MST Algorithms
- Lesson 5: Hands-On Practice
Chapter 20: Fundamental Theorems in Graph Theory
- Lesson 1: Kuratowski’s Theorem: Planarity and Graph Structure
- Lesson 2: Euler’s Theorem: Bridges of Königsberg Problem
- Lesson 3: Ramsey Theory: Graphs and Combinatorial Properties
- Lesson 4: Turan’s Theorem: Maximum Number of Edges in Graphs
Chapter 21: Graph Traversal Applications
- Lesson 1: DFS and BFS in Web Crawling
- Lesson 2: Social Network Analysis Using Graph Traversal
- Lesson 3: Applications of Graph Traversal in AI and Robotics
- Lesson 4: Shortest Path Problems: Real-Life Use Cases
Chapter 22: Introduction to Network Flow Algorithms
- Lesson 1: Basics of Network Flow: Definitions and Applications
- Lesson 2: Max-Flow Min-Cut Theorem: Understanding Its Importance
- Lesson 3: Ford-Fulkerson Algorithm: Maximum Flow Calculation
- Lesson 4: Applications of Network Flow in Logistics and Telecom
Chapter 23: Graph-Based Data Structures
- Lesson 1: Adjacency List vs. Adjacency Matrix: Pros and Cons
- Lesson 2: Graph Implementation in Python, Java, and C++
- Lesson 3: Using Graphs in Database Indexing
- Lesson 4: Applications of Graph-Based Data Structures in Big Data
Chapter 24: Graph Algorithms in Computational Biology
- Lesson 1: Graphs in Genomics: DNA and Protein Analysis
- Lesson 2: Sequence Alignment Using Graph Models
- Lesson 3: Biological Network Analysis: Protein-Protein Interactions
- Lesson 4: Evolutionary Trees and Phylogenetic Graphs
Chapter 1: Advanced Graph Properties and Metrics
- Lesson 1: Centrality Measures (Degree, Betweenness, Closeness, Eigenvector)
- Lesson 2: Graph Diameter and Radius
- Lesson 3: Graph Spectral Properties
- Lesson 4: Clustering Coefficient and Graph Density Analysis
- Lesson 5: Applications of Advanced Metrics
Chapter 2: Advanced Network Flow Algorithms
- Lesson 1: Advanced Max-Flow Techniques
- Lesson 2: Edmonds-Karp Algorithm
- Lesson 3: Push-Relabel Algorithm
- Lesson 4: Min-Cost Flow Problem
- Lesson 5: Real-World Flow Optimization Problems
Chapter 3: Graph Theory in Computational Complexity
- Lesson 1: Complexity Classes (P, NP, and NP-Complete Graph Problems)
- Lesson 2: Traveling Salesman Problem (TSP) and Approximation Algorithms
- Lesson 3: Graph Coloring in Computational Complexity
- Lesson 4: Independent Set and Vertex Cover Problems
- Lesson 5: Reductions Between Graph Problems
Chapter 4: Graph Partitioning and Cuts
- Lesson 1: Minimum Cut Problem
- Lesson 2: Graph Partitioning Algorithms
- Lesson 3: Applications in Distributed Computing and Network Design
- Lesson 4: Spectral Graph Partitioning
- Lesson 5: Case Studies in Partitioning
Chapter 5: Random Graphs and Probabilistic Graph Theory
- Lesson 1: Introduction to Random Graph Models
- Lesson 2: Erdos-Rényi Model: Random Graph Generation
- Lesson 3: Percolation Theory and Random Walks
- Lesson 4: Applications in Epidemic Modeling and Spread Prediction
- Lesson 5: Small-World Networks (Watts-Strogatz Model)
- Lesson 6: Scale-Free Networks (Barabási-Albert Model)
- Lesson 7: Applications of Random Graphs in Social and Biological Networks
- Lesson 8: The Role of Random Graphs in Machine Learning
Chapter 6: Advanced Graph Traversal Algorithms
- Lesson 1: Bidirectional Search
- Lesson 2: A* Search Algorithm
- Lesson 3: Iterative Deepening Search
- Lesson 4: Applications in AI and Robotics
- Lesson 5: Comparison of Advanced Traversal Algorithms
Chapter 7: Graph Embedding and Graph Neural Networks (GNNs)
- Lesson 1: Introduction to Graph Embedding
- Lesson 2: Node2Vec and DeepWalk Algorithms
- Lesson 3: Basics of Graph Neural Networks (GNNs)
- Lesson 4: Message Passing in GNNs: How They Learn
- Lesson 5: Applications of GNNs in AI and Machine Learning
- Lesson 6: Applications of GNNs in NLP
- Lesson 7: Implementing GNNs Using TensorFlow and PyTorch
- Lesson 8: Case Studies and Practical Implementations
Chapter 8: Graph Theory in Machine Learning
- Lesson 1: Graph-Based Clustering Techniques
- Lesson 2: Semi-Supervised Learning on Graphs
- Lesson 3: Graph Kernels for Classification
- Lesson 4: Applications in Recommendation Systems and Fraud Detection
- Lesson 5: Hands-On Implementation of Graph ML Models
Chapter 9: Dynamic Graphs and Temporal Graphs
- Lesson 1: What are Dynamic Graphs?
- Lesson 2: Temporal Graph Models
- Lesson 3: Algorithms for Updating Dynamic Graphs
- Lesson 4: Applications in Real-Time Network Analysis
- Lesson 5: Predicting Future Graph States Using AI
- Lesson 6: Case Studies
Chapter 10: Hypergraphs and Multilayer Networks
- Lesson 1: Introduction to Hypergraphs
- Lesson 2: Properties of Hypergraphs
- Lesson 3: Multilayer and Multiplex Networks
- Lesson 4: Applications in Biology and Social Sciences
- Lesson 5: Algorithms for Hypergraph Analysis
Chapter 11: Algebraic Graph Theory
- Lesson 1: Adjacency and Laplacian Matrices Revisited
- Lesson 2: Spectral Graph Theory
- Lesson 3: Graph Eigenvalues and Eigenvectors
- Lesson 4: Cheeger’s Inequality and Expander Graphs
- Lesson 5: Applications in Machine Learning and Network Science
- Lesson 6: Applications in Community Detection
- Lesson 7: Hands-On Spectral Graph Analysis
Chapter 12: Graph Theory in Optimization
- Lesson 1: Graph Theory in Linear and Integer Programming
- Lesson 2: Shortest Path Variants (k-Shortest Path, Constrained Shortest Path)
- Lesson 3: Multi-Objective Optimization in Graphs
- Lesson 4: Applications in Supply Chain and Logistics
- Lesson 5: Practical Optimization Problems
Chapter 13: Graph Theory in Parallel and Distributed Computing
- Lesson 1: Parallel Graph Algorithms
- Lesson 2: Distributed Graph Processing Frameworks (e.g., Apache Giraph, Pregel)
- Lesson 3: Scalability Challenges in Large-Scale Graphs
- Lesson 4: Case Studies in Parallel Computing
- Lesson 5: Graph Theory in Cloud Environments
Chapter 14: Graph Theory in Cryptography and Network Security
- Lesson 1: Graph-Based Cryptographic Models and Protocols
- Lesson 2: Network Security Using Graphs
- Lesson 3: Graph-Based Attack Detection and Threat Analysis in Cybersecurity
- Lesson 4: Zero-Knowledge Proofs Using Graph Theory
- Lesson 5: Applications in Blockchain and Secure Transactions
- Lesson 6: Case Studies in Cybersecurity
Chapter 15: Advanced Applications of Graph Theory
- Lesson 1: Graph Theory in Biology (Protein Interaction Networks)
- Lesson 2: Graph Theory in Chemistry (Molecular Graphs)
- Lesson 3: Graph Theory in Transportation and Urban Planning
- Lesson 4: Graph Theory in Social Network Analysis
- Lesson 5: Case Studies and Emerging Trends
Chapter 16: Advanced Graph Coloring
- Lesson 1: Interval Graphs, Chordal Graphs and Perfect Graph Theorems
- Lesson 2: Edge Coloring and Total Coloring
- Lesson 3: Algorithms for Advanced Coloring Problems
- Lesson 4: Applications in Frequency Assignment
- Lesson 5: Chromatic Polynomial: Counting Colorings
- Lesson 6: List Coloring and Precoloring Extension
- Lesson 7: Graph Coloring in Scheduling and Map Coloring
- Lesson 8: Open Problems in Graph Coloring
Chapter 17: Graph Isomorphism and Automorphism
- Lesson 1: Advances in Graph Isomorphism Testing
- Lesson 2: Practical Applications of Graph Automorphisms
- Lesson 3: Computational Complexity of Isomorphism Problems
- Lesson 4: Graph Symmetry and Automorphism Groups
- Lesson 5: Applications in Chemistry and Pattern Recognition
- Lesson 6: Recent Developments in the Field
Chapter 18: Research and Open Problems in Graph Theory
- Lesson 1: Overview of Current Research Topics
- Lesson 2: Conjectures in Graph Theory (e.g., Four Color Theorem)
- Lesson 3: Open Problems in Spectral Graph Theory
- Lesson 4: Collaboration and Resources for Graph Theory Research
- Lesson 5: Future Directions
Chapter 19: Tools and Libraries for Advanced Graph Analysis
- Lesson 1: Overview of Popular Libraries (NetworkX, igraph, SNAP)
- Lesson 2: Using Neo4j for Graph Databases
- Lesson 3: Advanced Visualization Tools (Gephi, Cytoscape)
- Lesson 4: Parallel Graph Processing Libraries
- Lesson 5: Hands-On Projects with Tools
Chapter 20: NetworkX: Graph Analysis in Python
- Lesson 1: Introduction to NetworkX: Features and Installation
- Lesson 2: Creating Graphs: Undirected, Directed, and Multigraphs
- Lesson 3: Adding Nodes and Edges: Attributes and Weights
- Lesson 4: Graph Traversal and Search Algorithms (BFS, DFS)
- Lesson 5: Shortest Path Algorithms: Dijkstra and Bellman-Ford
- Lesson 6: Network Centrality Measures: Degree, Betweenness, Closeness
- Lesson 7: Community Detection Using NetworkX
- Lesson 8: Graph Visualization in NetworkX
- Lesson 9: Real-World Applications of NetworkX in Social Networks and AI
- Lesson 10: Case Study: Analyzing a Social Network Graph
Chapter 21: Gephi: Visualizing Graph Data
- Lesson 1: Introduction to Gephi: Features and Installation
- Lesson 2: Importing Graph Data: Supported Formats and Parsing
- Lesson 3: Exploring Graph Layouts: Force-Directed and Hierarchical
- Lesson 4: Graph Filtering and Clustering: Working with Large Graphs
- Lesson 5: Graph Metrics and Network Analysis in Gephi
- Lesson 6: Dynamic Graph Visualization: Time-Based Changes
- Lesson 7: Exporting and Sharing Graph Visualizations
- Lesson 8: Using Gephi for Social Network Analysis
- Lesson 9: Gephi in Scientific Research and Business Intelligence
- Lesson 10: Case Study: Visualizing a Citation Network
Chapter 22: igraph: Advanced Graph Analysis
- Lesson 1: Introduction to igraph: Features and Installation
- Lesson 2: Creating and Manipulating Graphs in Python, R, and C
- Lesson 3: Graph Data Structures: Adjacency Lists and Matrices
- Lesson 4: Shortest Path Algorithms: Comparing Efficiency
- Lesson 5: Community Detection with Modularity and Louvain Algorithm
- Lesson 6: Graph Randomization and Statistical Network Analysis
- Lesson 7: Using igraph for Large-Scale Graph Processing
- Lesson 8: Visualization of Graphs with igraph
- Lesson 9: Applications in Bioinformatics and Computational Social Science
- Lesson 10: Case Study: Modeling a Citation Network with igraph
Chapter 23: Graphviz: Graph Visualization with DOT Language
- Lesson 1: Introduction to Graphviz: Overview and Installation
- Lesson 2: Understanding DOT Language: Syntax and Structure
- Lesson 3: Creating Simple Graphs and Customizing Appearance
- Lesson 4: Directed vs. Undirected Graphs in Graphviz
- Lesson 5: Applying Graph Layout Algorithms: Hierarchical, Radial, Circular
- Lesson 6: Adding Labels, Colors, and Styles to Graphs
- Lesson 7: Exporting Graphs to Different Formats (PNG, SVG, PDF)
- Lesson 8: Automating Graph Generation Using Python and Graphviz
- Lesson 9: Graphviz Applications in Software Engineering and System Design
- Lesson 10: Case Study: Visualizing an Organizational Hierarchy
Chapter 24: SageMath: Graph Theory Computations
- Lesson 1: Introduction to SageMath: Features and Installation
- Lesson 2: Working with Graph Data Structures in SageMath
- Lesson 3: Graph Traversal Algorithms: BFS, DFS, and Variants
- Lesson 4: Computing Shortest Paths and Minimum Spanning Trees
- Lesson 5: Graph Theoretic Properties: Cliques, Cycles, and Components
- Lesson 6: Eigenvalues and Spectral Graph Theory in SageMath
- Lesson 7: Using SageMath for Graph Automorphism and Isomorphism Tests
- Lesson 8: Custom Graph Generation and Visualization
- Lesson 9: Applications in Mathematics, Physics, and Cryptography
- Lesson 10: Case Study: Modeling a Road Network with SageMath
Chapter 25: Extremal Graph Theory and Applications
- Lesson 1: Introduction to Extremal Graph Theory
- Lesson 2: Erdős–Stone Theorem: Upper Bounds on Graphs
- Lesson 3: Forbidden Subgraphs and Applications
- Lesson 4: Applications in Algorithmic Design
Chapter 26: Planarity and Dual Graphs
- Lesson 1: Characterization of Planar Graphs
- Lesson 2: Kuratowski’s Theorem: Planarity Testing
- Lesson 3: Dual Graphs and Their Properties
- Lesson 4: Applications of Planar Graphs in Circuit Design
Chapter 27: Advanced Network Flow Algorithms
- Lesson 1: Push-Relabel Algorithm: Maximum Flow Computation
- Lesson 2: Dinic’s Algorithm: Faster Flow Computation
- Lesson 3: Applications in Traffic and Internet Routing
- Lesson 4: Min-Cost Flow Problems and Their Solutions
Chapter 28: Graph-Based Clustering Techniques
- Lesson 1: Spectral Clustering: Using Eigenvalues for Clustering
- Lesson 2: Community Detection in Social Networks
- Lesson 3: Graph Laplacians and Their Role in Clustering
- Lesson 4: Applications in Image Segmentation
Chapter 29: Graph Theory in Quantum Computing
- Lesson 1: Quantum Graph Algorithms: Basics and Fundamentals
- Lesson 2: Graph-Based Quantum Error Correction Codes
- Lesson 3: Applications in Quantum Cryptography
- Lesson 4: Graph Representation in Quantum Networks
Chapter 30: Large-Scale Graph Processing
- Lesson 1: Distributed Graph Algorithms for Big Data
- Lesson 2: Graph Processing Frameworks: Pregel, GraphX, and Neo4j
- Lesson 3: Scaling Graph Algorithms for Large Networks
- Lesson 4: Real-World Applications in Social Media and Web Search
Chapter 31: Multilayer and Multiplex Networks
- Lesson 1: Multilayer Networks: Interconnected Graphs in Reality
- Lesson 2: Modeling Interdependent Systems Using Multiplex Graphs
- Lesson 3: Real-World Applications in Biology and Sociology
- Lesson 4: Multilayer Graph Visualization Techniques
Chapter 32: Capstone Project and Final Thoughts
- Lesson 1: Designing a Research-Oriented Capstone Project
- Lesson 2: Implementing Graph Algorithms in Practice
- Lesson 3: Interdisciplinary Applications of Graph Theory
- Lesson 4: Recap of Advanced Topics
- Lesson 5: Final Q&A and Resources for Further Learning
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