Discrete Mathematics - Vol 3 | 5. Lecture - 54 by Abraham | Learn Smarter
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5. Lecture - 54

The chapter delves into the concept of graphic sequences in graph theory, specifically focusing on the Havel-Hakimi theorem for determining if a given degree sequence can represent a simple graph. It outlines necessary conditions for a sequence to be classified as graphic, offers methods for constructing sequences, and provides detailed proofs of the theorem's implications. Additionally, the chapter presents exercises and activities that reinforce the concepts discussed throughout.

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Sections

  • 5.1

    Discrete Mathematics

    This section introduces the concept of degree sequences in graphs and the conditions for a sequence to be classified as graphic.

    Learning Practice

  • 1.1

    Lecture - 54

    This lecture focuses on the concept of graphic sequences in graph theory, particularly the Havel-Hakimi theorem, and how to determine if a given degree sequence can form a simple graph.

    Learning Practice

  • 5.1.2

    Tutorial 9: Part Ii

    This section discusses the properties of graphic sequences in graph theory and introduces the Havel-Hakimi theorem as a method for characterizing graphic sequences.

    Learning Practice

  • 5.2

    Degree Sequence Of A Graph

    The degree sequence of a graph refers to the non-increasing order list of vertex degrees, with the section delving into the conditions under which a sequence can be classified as a graphic sequence.

    Learning Practice

  • 5.2.1

    Definition

    This section defines the degree sequence of a graph and explores the concept of graphic sequences.

    Learning Practice

  • 5.2.2

    Questions

    This section discusses degree sequences in graphs and introduces the Havel-Hakimi theorem, which provides a method for determining if a sequence of integers can represent the degree sequence of a simple graph.

    Learning Practice

  • 5.2.3

    Example Sequences

    This section defines and explores degree sequences in graphs, focusing on graphic sequences and the Havel-Hakimi theorem.

    Learning Practice

  • 5.3

    Characterization Of Graphic Sequences

    This section introduces the concept of a degree sequence in graphs and the characterization of graphic sequences through the Havel-Hakimi theorem.

    Learning Practice

  • 5.3.1

    Havel-Hakimi Theorem

    The Havel-Hakimi theorem provides a necessary and sufficient condition to determine if a degree sequence can represent a simple graph.

    Learning Practice

  • 5.3.2

    Construction Of Sequence S*

    This section discusses the concept of graphic sequences and introduces the Havel-Hakimi theorem, which provides a method for determining whether a given degree sequence can correspond to a simple graph.

    Learning Practice

  • 5.3.3

    Verification Of Graphic Sequence

    This section introduces the concept of degree sequences in graphs and the criteria to verify if a sequence is graphic using the Havel-Hakimi theorem.

    Learning Practice

  • 5.4

    Proof Of The Havel-Hakimi Theorem

    The Havel-Hakimi Theorem provides a method to determine if a given sequence of integers can represent the degree sequence of a simple graph.

    Learning Practice

  • 5.4.1

    Implication One

    This section discusses the concept of degree sequences in graphs and the application of the Havel-Hakimi theorem to determine graphic sequences.

    Learning Practice

  • 5.4.2

    Implication Two

    This section discusses the concept of graphic sequences in graph theory, particularly focusing on the Havel-Hakimi theorem and its applications to verify whether a given sequence of vertex degrees corresponds to a simple graph.

    Learning Practice

  • 5.4.2.1

    Case 1: Vertex V Is Adjacent

    This section discusses the concept of graphic sequences in graph theory, specifically focusing on the degree sequence and the conditions under which they can form a simple graph.

    Learning Practice

  • 5.4.2.2

    Case 2: Vertex V Is Not Adjacent

    This section explores the concept of graphic sequences in graphs, particularly focusing on the necessary conditions for a sequence of vertex degrees to represent a simple graph.

    Learning Practice

References

ch55 - part A.pdf

Class Notes

Memorization

What we have learnt

  • A graphic sequence is one t...
  • The Havel-Hakimi theorem pr...
  • The sum of degrees in any g...

Final Test

Revision Tests