Graph Theory

  1. Fundamental Concepts and Types of Graphs
    1. Basic Terminology
      1. Adjacency
        1. Adjacent Vertices
          1. Adjacency in Undirected Graphs
            1. Adjacency in Directed Graphs
            2. Incidence
              1. Incident Edges and Vertices
                1. Incidence Relations
                2. Degree of a Vertex
                  1. Degree in Undirected Graphs
                    1. In-degree in Directed Graphs
                      1. Out-degree in Directed Graphs
                        1. Degree Sequence
                        2. Handshaking Lemma
                          1. Statement and Proof
                            1. Consequences for Graph Structure
                          2. Types of Graphs
                            1. Undirected Graphs
                              1. Definition and Properties
                                1. Edge Representation
                                2. Directed Graphs (Digraphs)
                                  1. Definition and Properties
                                    1. Arc Representation
                                      1. Underlying Undirected Graph
                                      2. Simple Graphs
                                        1. No Loops or Multiple Edges
                                        2. Multigraphs
                                          1. Multiple Edges Between Vertices
                                          2. Pseudographs
                                            1. Loops and Multiple Edges
                                              1. General Graph Structure
                                              2. Weighted Graphs
                                                1. Edge Weights
                                                  1. Weight Functions
                                                  2. Unweighted Graphs
                                                    1. Binary Edge Relations
                                                      1. Structural Focus
                                                    2. Special Graph Structures
                                                      1. Complete Graphs
                                                        1. Definition and Notation (Kn)
                                                          1. Number of Edges
                                                          2. Bipartite Graphs
                                                            1. Definition and Properties
                                                              1. Vertex Partitions
                                                                1. Characterization by Odd Cycles
                                                                  1. Testing for Bipartiteness
                                                                  2. Complete Bipartite Graphs
                                                                    1. Definition and Notation (Km,n)
                                                                    2. Cycles
                                                                      1. Definition and Notation (Cn)
                                                                        1. Cycle Length
                                                                        2. Paths
                                                                          1. Definition and Notation (Pn)
                                                                            1. Path Length
                                                                            2. Wheels
                                                                              1. Definition and Construction
                                                                                1. Hub and Rim Structure
                                                                                2. Regular Graphs
                                                                                  1. k-Regular Graphs
                                                                                  2. Petersen Graph
                                                                                    1. Construction
                                                                                      1. Significance in Graph Theory
                                                                                    2. Graph Isomorphism
                                                                                      1. Definition of Isomorphic Graphs
                                                                                        1. Formal Definition
                                                                                          1. Bijective Mapping
                                                                                            1. Structure Preservation
                                                                                              1. Examples and Non-examples
                                                                                              2. Invariants for Proving Non-Isomorphism
                                                                                                1. Degree Sequence
                                                                                                  1. Number of Vertices and Edges
                                                                                                    1. Number of Cycles
                                                                                                      1. Connectivity Properties
                                                                                                        1. Diameter
                                                                                                          1. Girth
                                                                                                            1. Other Structural Properties

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                                                                                                        1. Introduction to Graph Theory

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                                                                                                        3. Representing Graphs