5.4.1. Implication One

Try these first

1. 1.

   What does the term 'degree sequence' refer to in graph theory?

   Hint

   Think about what degree means for a vertex.

2. 2.

   True or False: A vertex can have a negative degree.

   Hint

   Consider what a degree represents.

3. 3.

   What is the degree sequence of a graph?

   • A list of the edges in ascending order
   • The list of edge weights
   • The list of vertices' degrees in non-increasing order

   Hint

   Consider how you list the degrees.

4. 4.

   True or False: All sequences that contain non-negative integers can be graphic.

   • True
   • False

   Hint

   Reflect on examples that couldn't satisfy graphical conditions.

5. 5.

   Construct a graphic sequence of your own with five vertices and explain why it is graphic.

   Hint

   Think about the connections you can make that meet the degree needs.

6. 6.

   Apply the Havel-Hakimi theorem to the sequence (8, 6, 6, 4, 2, 1). Show all steps to verify its graphic nature.

   Hint

   Follow the reduction process diligently and ensure you count correctly.