Pathway to Learning and Research in Graph Theory

[mak52810]

Hi All

As the title suggests I want to get to a level where I can approach research problems in Graph Theory. Specifically in the areas of Algebraic Graph Theory and Random Graphs. I know this is no small endeavour but I atleast want to put some direction into my extra studying.


My background in Mathematics is

-a year long sequence in Calculus using a text by Anton which I supplemented with Spivak

-a proof based course in Linear Algebra which used Leon which I supplemented with Hoffman and Kunze (I intend to cover what is left using Lang's Linear Algebra text)

-some work in Group Theory upto the first seven sections of A First Course in Abstract Algebra by JB Fraleigh. These days I am making up some ground in Algebra by watching the video lectures of Benedict Gross from Harvard and working through the associated parts of Artin's Algebra.

-A course in Discrete Mathematics using Rosen that covered logic, mathematical proofs, number theory and linear congruences. I was hoping that we would get some introduction to graphs and combinatorics but time did not permit it.

-Three courses in Statistics and Probability.

So my questions are:

-how much algebra/linear algebra should I cover?

-What books should I use to introduce myself to Graph Theory? Currently I have Introduction to Graph Theory by Trudeau and Introductory Graph Theory by Chartrand. Getting new books isn't a problem I can get them through my university library or get them ordered if they aren't there.

-After that what books should I use to get myself well acquainted with advanced areas in Graph Theory such as Algebraic Graph Theory and Random Graphs?


Looking forward to replies.
Thank you in advance.

 

[TonyG]

Thank you for your interest in Graph Theory and your dedication to further your knowledge in this field. Based on your background, it seems like you have a strong foundation in mathematics, which is essential for approaching research problems in Graph Theory.

To answer your first question, I would recommend covering a good amount of algebra and linear algebra before delving into advanced areas of Graph Theory. This will help you understand the underlying algebraic structures and concepts that are often used in Graph Theory. I would suggest studying topics such as group theory, ring theory, and linear algebra in depth, as they will be useful for understanding topics in Algebraic Graph Theory and Random Graphs.

For your second question, Introduction to Graph Theory by Trudeau and Introductory Graph Theory by Chartrand are both good introductory books to get you started. Another book that I would recommend is Graph Theory by Bondy and Murty. It provides a comprehensive introduction to the subject and covers many important topics in Graph Theory.

For advanced areas such as Algebraic Graph Theory and Random Graphs, I would suggest the following books:

1. Algebraic Graph Theory by Norman Biggs
2. Random Graphs by Béla Bollobás
3. Graph Theory and Complex Networks by Maarten van Steen

These books are widely used in academia and will provide you with a solid understanding of these topics.

In addition to these books, I would also recommend attending conferences and workshops related to Graph Theory, as well as reading research papers in the field. This will help you stay updated on current research and also give you the opportunity to network with other researchers and experts in the field.

I wish you all the best in your studies and research in Graph Theory. Keep up the hard work and dedication, and I have no doubt that you will achieve your goal of approaching research problems in this field. Good luck!