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Introduction to Graph Theory (Dover Books on Mathematics)
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Well I'm not a mathematician so I'm not sure my advice is going to be useful.
If you're eager and enjoy a challenge I'd say my one regret was not learning how to construct my own proofs until much later on. Learning how to apply maths to solve problems is a lot of fun but learning how to think abstractly and make your own arguments is much more satisfying. There's a great book that doesn't require too much more than HS level math to understand which starts to make this connection called, Introduction To Graph Theory  and it's one of my all-time favorites.
Sort of. That would absolutely be "allowed". Now whether that is useful to write out, depends on what problem you are trying to solve.
Here is another example
> Does the book cover all the relevant parts
No, it isn't quite so comprehensive, but it will absolutely help you get started and help you decide if you want to learn more.
Read some books, practice exercises, and find an area of interest.
Start with some liberal-arts introduction to a particular topic of interest and delve in.
I often find myself recommending Introduction to Graph Theory . It is primarily aimed at liberal arts people who are math curious but may have been damaged or put off by the typical pedagogy of western mathematics. It will start you off by introducing some basic material and have you writing proofs in a simplistic style early on. I find the idea of convincing yourself it works is a better approach to teaching than to simply memorize formulas.
Another thing to ask yourself is, what will I gain from this? Mathematics requires a sustained focus and long-term practice. Part of it is rote memorization. It helps to maintain your motivation if you have a reason, a driving reason, to continue this practice. Even if it's simply a love of mathematics itself.
For me it was graphics at first... and today it's formal proofs and type theory.
Mathematics is beautiful. I'm glad we have it.
Update: I also recommend keeping a journal of your progress. It will be helpful to revisit later when you begin to forget older topics and will help you to create a system for keeping your knowledge fresh as you progress to more advanced topics.
Graph theory is amazing. One of my favorite subjects.
If you're a liberal-arts kind of person and or have scars from prior experiences with mathematics I recommend, https://www.amazon.com/Introduction-Graph-Theory-Dover-Mathe...
A more graduate-level book but one I found pleasing:
A Logical Approach to Discrete Mathematics: https://www.amazon.com/Logical-Approach-Discrete-Monographs-...
And a more pragmatic approach to the same material (with a lot of cross-over in terms of proof-style, etc):
Programming in the 1990s: http://www.springer.com/gp/book/9780387973821
But one I particularly enjoyed early on was written for liberal-arts level students of maths (who might've been traumatized by maths in the past):
Introduction to Graph Theory: https://www.amazon.com/Introduction-Graph-Theory-Dover-Mathe...
It will actually get you into writing proofs in set theory within the first couple of chapters.
⬐ bordercasesOh gosh the equational logic rabbit hole.
To add to the fire: http://mathmeth.com/
Here is a great one: http://www.amazon.com/Introduction-Graph-Theory-Dover-Mathem...
"A stimulating excursion into pure mathematics aimed at "the mathematically traumatized," but great fun for mathematical hobbyists and serious mathematicians as well. Requiring only high school algebra as mathematical background, the book leads the reader from simple graphs through planar graphs, Euler's formula, Platonic graphs, coloring, the genus of a graph, Euler walks, Hamilton walks, and a discussion of The Seven Bridges of Konigsberg."
There's also stuff under "Network theory" at Wikipedia. I feel like those two articles should probably be merged, but it hasn't happened yet, and I haven't had time to take a stab at it. But anyway, both articles contain some useful info.
I also recommend these few books as a good starting point:
Network Science: Theory and Applications
Linked: How Everything Is Connected to Everything Else and What It Means
Six Degrees: The Science of a Connected Age
The Wisdom of Crowds
Nexus: Small Worlds and the Groundbreaking Science of Networks
Diffusion of Innovations
Of course - being that Network Science is a multidisciplinary field, that touches a lot of other areas - it can be hard to get a handle on what to study. But those few books - between them - cover a lot of the basics and would give somebody who's interested in this stuff enough background to figure out where to start digging deeper.
For a little bit more on the technical side, a couple of good resources at:
Introductory Graph Theory
Introduction to Graph Theory
Algorithms in Java: Part 5 - Graph Algorithms
⬐ mysterywhiteboyI would add near the top of your list the awesome (and free) book by David Easley and Jon Kleinberg that accompanies their Cornell undergraduate course:
Networks, Crowds, and Markets: Reasoning About a Highly Connected World.⬐ mindcrimeOooh, good call. I hadn't read that one, but it looks very good.
I like the cheap Dover intro: http://www.amazon.com/Introduction-Graph-Theory-Advanced-Mat...
⬐ antesterIf this is the same Dover book we used in our graph theory class, I found it difficult to learn from without a professor explaining the unclear sections. Not just doing the exercises, but some of the things they presented as trivial had some important steps missing.