Find an Interesting Topic in Graph Theory for an Exam

In summary, Graph Theory is a branch of mathematics that deals with the study of graphs, which are structures that represent relationships between objects. Some interesting topics in this field that could be explored in an exam include the Four Color Theorem, which states that any map can be colored using only four colors without any two adjacent regions sharing the same color, and the Travelling Salesman Problem, which involves finding the shortest possible route for a salesman to visit all given cities and return to the starting point. Other topics such as connectivity, planarity, and network flow also offer intriguing challenges for students studying Graph Theory.
  • #1
BRN
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Homework Statement
Topic concerning graph theory
Relevant Equations
None.
Hello everybody,

I hope it is the right section to post.

For an exam, I should delve into a topic concerning graph theory. My work should include theoretical explanation, pseudo code, correctness analysis, complexity analysis and code implementation (C ++, python or other).

Could someone recommend me an interesting topic to discuss?

Thanks so much!
 
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  • #2
First here are some practice problems that might give you an idea for a more original one:

https://www.geeksforgeeks.org/graph-theory-practice-questions/

and this one:

https://towardsdatascience.com/common-graph-theory-problems-ca990c6865f1

Next a list of open problems in graph theory (the latest unsolved problems) which may be a bit beyond your skill:

http://openproblemgarden.org/category/graph_theory

where you might find a numerical solution but yet not be able to prove some conjecture (unless you can find a counter example). Anyway, they are at least worth looking at.
 
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  • #3
Thank you so much!

The last link is very interesting!
 
  • #4
An interesting theoretical topic would be a program that reduces one NP-complete problem in graph theory to another one. See Karp reduction.
 
  • #5
This tells of early work by Euler establishing foundations of graph theory and topology:
https://en.wikipedia.org/wiki/Seven_Bridges_of_Königsberg

This introduces the Chinese Postman Problem, and the Travelling Salesman Problem, or Salesman's Route Problem (an archetype of NP-complete problems): https://en.wikipedia.org/wiki/Route_inspection_problem

This is a linear-time algorithm for finding a minimum-weight spanning tree: https://en.wikipedia.org/wiki/Prim's_algorithm

And planarity testing is important in the design of micro-circuitry:
https://en.wikipedia.org/wiki/Planarity_testing
https://www.eeeguide.com/methods-of-circuit-analysis/

For some background on the history of implementation of planarity in integrated circuits, before the importance of that became generally appreciated, please look at:
https://www.edn.com/noyce-receives-1st-ic-patent-april-25-1961/
https://www.edn.com/noyce-conceives-planar-ic-january-23-1959/
 
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  • #6
A topical area would be networks in epidemiology.
 
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1. What is graph theory?

Graph theory is a branch of mathematics that deals with the study of graphs, which are mathematical structures used to represent relationships between objects. It involves the analysis of the properties and behavior of graphs, as well as the development of algorithms and models to solve problems related to them.

2. Why is graph theory an important topic to study?

Graph theory has a wide range of applications in various fields such as computer science, engineering, social sciences, and biology. It provides a powerful framework for solving real-world problems that involve networks or relationships between entities.

3. How can I find an interesting topic in graph theory for an exam?

One way to find an interesting topic in graph theory is to look for current research trends and recent developments in the field. You can also consider real-world problems that can be modeled and solved using graph theory, or explore the applications of graph theory in different areas.

4. What are some examples of interesting topics in graph theory?

Some examples of interesting topics in graph theory include network analysis, graph coloring, shortest path algorithms, social network analysis, and graph embeddings. Other topics of interest may include graph clustering, graph theory in machine learning, and graph visualization techniques.

5. How can I prepare for an exam on graph theory?

To prepare for an exam on graph theory, it is important to have a good understanding of the fundamental concepts and principles of the subject. Practice solving problems and working on sample questions to improve your problem-solving skills. You can also review lecture notes, textbooks, and online resources to reinforce your knowledge and understanding of the topic.

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