Graph Theory Tutorial

Graph Theory is a branch of mathematics concerned with the study of objects (called vertices or nodes) and the connections between them (called edges). A graph is a collection of various vertices, also known as nodes, and these nodes are connected via edges.

Graphs are widely used in computer science, network analysis, operations research, and real-world problem solving.

Basics of Graph Theory

Covers the foundations of graphs, their representations, key terminology, and basic algorithms like Dijkstra’s.

• Introduction to Graph Theory
• Graph Representation
• Walks, Trails, Paths, and Circuits
• Graph Measurements
• Cut-Vertices and Cut-Edges
• Bridge in Graph
• Independent Sets
• Dijkstra's Algorithm
• Application of Graph Theory

Graph Traversals

Learn how to explore graphs systematically using DFS, BFS, and topological sorting.

• Depth-First Search [DFS]
• Breadth-First Search [BFS]
• Topological Sorting

Trees

Focuses on hierarchical graph structures, spanning trees, traversals, and coding applications.

• Introduction to Trees
• Prim's Minimum Spanning Tree 
• Kruskal's Minimum Spanning Tree
• Huffman Codes 
• Tree Traversals
• Traveling Salesman Problem
• Binary Trees and Binary Search Trees
• AVL Tree and Balanced Tree

Special Graphs

Introduces important classes of graphs like bipartite, complete, regular, and random graphs.

• Bipartite Graphs
• Regular Graphs
• Complete Graph
• Erdos Renyi Model (Random Graph)
• Independent Sets and Covering

Eulerian Graphs

Study Eulerian paths and cycles, along with algorithms and real-world applications.

• Introduction to Eulerian Graphs
• Fleury’s Algorithm
• Chinese-Postman-Problem Hamilton

Matching

Learn about matching problems, including perfect and bipartite matchings, with approximation methods.

• Matching- Basics, Perfect, Bipartite
• Approximation Algorithms

Coloring

Understand graph coloring, chromatic numbers, and their applications in scheduling and optimization.

• Introduction to Graph Coloring
• Chromatic Numbers
• Greedy Coloring Algorithm
• Edge Coloring

Planar Graph

Explores planar graphs, planarity testing, and Kuratowski’s characterization.

• Introduction to Planar Graph
• Kuratowskis Graphs

Directed Graphs

Covers properties of digraphs, connectivity, shortest paths, and strong components.

• Degree Centrality
• Weak Connectivity
• Strong Components 
• Eulerian, Hamiltonian Directed Graphs
• Tarjan's Algorithm
• Bellman-Ford Algorithm
• Floyd-Warshall Algorithm
• Handshaking in Graph Theorem

Group Theory

Links algebraic structures with graph theory through groups, isomorphisms, and fields.

• Groups, Subgroups, Semi-Groups
• Isomorphism, Homomorphism
• Automorphism
• Rings, Integral Domains, Fields