Discrete Mathematics - Vol 2 | 24. Graph Theory Basics by Abraham | Learn Smarter
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24. Graph Theory Basics

Graph theory encompasses a vast realm of concepts involving vertices and edges, including specialized structures like complete graphs, bipartite graphs, and cycle graphs. Fundamental theorems such as the Handshaking theorem and Euler's theorem provide insight into the properties of graphs, particularly regarding vertex degrees and connectivity. Understanding these concepts establishes a foundation for exploring more complex topics in graph theory.

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Sections

  • 24.1

    Graph Theory Basics

    This section introduces the fundamental concepts of graph theory, including definitions, types of graphs, and key theorems.

    Learning Practice

  • 24.1.1

    Definition Of A Graph

    A graph is defined as a collection of vertices and edges, with various types of graphs such as directed and undirected.

    Learning Practice

  • 24.1.2

    Types Of Graphs

    This section introduces various types of graphs in graph theory, including directed and undirected graphs, simple graphs, and special types like complete graphs and bipartite graphs.

    Learning Practice

  • 24.1.3

    Simple Graph Definition

    A simple graph consists of a collection of vertices and edges with no self-loops and at most one edge between any two vertices.

    Learning Practice

  • 24.1.4

    Terminologies Related To Undirected Graphs

    This section introduces fundamental terminologies related to undirected graphs, including definitions of graphs, simple graphs, degrees of vertices, and special types of undirected graphs.

    Learning Practice

  • 24.1.5

    Degree Of A Vertex

    This section defines the degree of a vertex in graph theory and explores its implications, including concepts like adjacency and the Handshaking theorem.

    Learning Practice

  • 24.1.6

    Handshaking Theorem

    The Handshaking Theorem states that in any undirected graph, the sum of the degrees of all vertices equals twice the number of edges.

    Learning Practice

  • 24.1.7

    Euler's Theorem

    Euler's Theorem states that in an undirected graph, the number of vertices with odd degree is always even.

    Learning Practice

  • 24.1.8

    Special Types Of Undirected Graphs

    This section introduces special types of undirected graphs, elaborating on their key properties and classifications.

    Learning Practice

  • 24.1.8.1

    Complete Graph

    The section introduces the concept of a complete graph, defining its properties and significance in graph theory.

    Learning Practice

  • 24.1.8.2

    Cycle Graph

    In this section, we explore cycle graphs, a specific type of simple graph where vertices are connected in a closed loop.

    Learning Practice

  • 24.1.8.3

    Wheel Graph

    This section discusses the concept of wheel graphs, a specific type of graph formed by adding a central vertex to a cycle graph.

    Learning Practice

  • 24.1.8.4

    Hypercube Graph (Q-N)

    The section introduces the concept of hypercube graphs, which consist of nodes representing n-bit strings, showing how edges connect nodes that differ by exactly one bit.

    Learning Practice

  • 24.1.9

    Bipartite Graphs

    Bipartite graphs consist of two disjoint sets of vertices, such that edges connect vertices from different sets only.

    Learning Practice

  • 24.1.10

    Complete Bipartite Graph

    A complete bipartite graph is a special type of bipartite graph where every vertex in one set is connected to every vertex in the other set, ensuring full connectivity.

    Learning Practice

  • 24.2

    Lecture Conclusion And References

    This section summarizes key concepts covered in the lecture on graph theory and provides references for further reading.

    Learning Practice

References

ch45.pdf

Class Notes

Memorization

What we have learnt

  • Graphs consist of vertices ...
  • Bipartite graphs require a ...
  • Euler's theorem indicates t...

Final Test

Revision Tests