Discrete Mathematics

1. Graph Theory
   1. Basic Definitions
      1. Graphs and Digraphs
         1. Definition of Graphs
            1. Vertices (Nodes)
            2. Edges (Arcs)
         2. Directed Graphs (Digraphs)
            1. Directed Edges
            2. In-degree and Out-degree
      2. Adjacency and Incidence
         1. Adjacency List
         2. Adjacency Matrix
         3. Incidence Matrix
         4. Edge List
   2. Types of Graphs
      1. Undirected and Directed Graphs
         1. Simple Graphs
         2. Multigraphs
         3. Pseudographs
      2. Weighted Graphs
         1. Representation of Weighted Graphs
         2. Applications in Real-world Problems
      3. Bipartite Graphs
         1. Properties of Bipartite Graphs
         2. Bipartite Matching
         3. Kőnig's Theorem
      4. Planar Graphs
         1. Euler's Formula
         2. Kuratowski's Theorem
         3. Face and Dual Concepts
      5. Trees and Forests
         1. Properties of Trees
         2. Binary Trees
         3. Tree Traversals
         4. Minimum Spanning Trees
   3. Graph Properties and Metrics
      1. Degree of Vertex
         1. Handshaking Lemma
         2. Degree Sequence
      2. Path and Cycle
         1. Eulerian Path and Circuit
         2. Hamiltonian Path and Cycle
         3. Dirac's and Ore's Theorem
      3. Connectivity
         1. Components
         2. Connected Graphs
         3. Strong and Weak Connectivity in Directed Graphs
         4. Cut Vertices and Bridges
         5. Menger's Theorem
   4. Graph Algorithms
      1. Shortest Path Algorithms
         1. Dijkstra's Algorithm
         2. Bellman-Ford Algorithm
         3. Floyd-Warshall Algorithm
      2. Minimum Spanning Tree
         1. Prim's Algorithm
         2. Kruskal's Algorithm
         3. Borůvka's Algorithm
      3. Network Flow Algorithms
         1. Maximum Flow Problem
         2. Ford-Fulkerson Algorithm
         3. Edmonds-Karp Algorithm
         4. Applications in Dinic's and Push-Relabel Algorithm
   5. Graph Colouring
      1. Chromatic Number
         1. Greedy Colouring Algorithm
         2. Five Colour Theorem
         3. Four Colour Theorem
      2. Chromatic Polynomials
         1. Calculation of Chromatic Polynomials
         2. Applications and Properties
      3. Vertex Colouring
         1. Edge Colouring
         2. Total Colouring
         3. Applications in Scheduling Problems