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I am an undergraduate student who hoping to self teach Graph Theory. I have studied elementary graph theory before, and have recently started reading 'Bollobas - Modern Graph Theory'. What are your recommendations after this? I have heard that topology and combinatorics are related to graph theory. What should I study alongside Graph Theory to gain deeper understanding of it? I am not interested in applications to computer science by the way. And textbook recommendations are also welcomed.

Thank you.

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    $\begingroup$ Matroid theory is a nice extension of many aspects of graph theory. But probability theory is probably the most important subject to know for modern graph theory, (much?) more than topology. $\endgroup$
    – Casteels
    Commented Jun 10, 2015 at 20:30
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    $\begingroup$ Here is a similar question, where I gave an answer that might be useful to you: math.stackexchange.com/questions/1307697/… $\endgroup$
    – Perry Elliott-Iverson
    Commented Jun 11, 2015 at 17:05

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Regarding your request for textbook recommendations: check them out at Amazon or your local library

Alon, N., The Probabilistic Method, John Wiley & Sons, 3rd Edition, 2008

Biggs, N., Algebraic Graph Theory, Cambridge University Press, 2nd Edition, 1996

Bondy, J. A., Graph Theory, Springer, 2008

Capobianco, M., Examples and Counterexamples in Graph Theory, Elsevier North-Holland, 1978

Diestel, R., Graph Theory, Springer, 4th Edition, 2012

Godsil, C., Algebraic Graph Theory, Springer, 2001

Graham, R. L., Concrete Mathematics, Addison-Wesley, 3nd Edition, 1994

Hartsfield, N., Pearls in Graph Theory, Dover, 2003

Jensen, T. R., Graph Coloring Problems, John Wiley \& Sons, 1995

Lovasz, L., Matching Theory, AMS, 2009

Lovasz, L., Combinatorial Problems and Exercises, AMS, 4nd Edition, 2007

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Several things, including:

  • Ramsey theory
  • Algebraic Graph theory and its applications in group theory
  • Spectral Graph theory and its applications in linear algebra
  • Knot theory and topological graph theory and its applications in topology.
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I would recommend the book Graphs and Digraphs 5th edition by Zhang, Lesniak, and Chartrand. It does a good job of covering many different aspects of graph theory: use of matrices (matrix-tree theorem, notably), topological graph theory (starts with the ubiquitous study of planar graphs and then moves to embedding graphs in surfaces of higher genus), Networks, and extremal graph theory and Ramsey theory (these are way at the end of the book as it takes a good deal of prior knowledge to study these things). Largely, it does not go into things like enumerative graph theory or random graph theory, nor does it go into hyper graph theory or matroid theory.

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