Best graph theory books according to redditors
We found 82 Reddit comments discussing the best graph theory books. We ranked the 31 resulting products by number of redditors who mentioned them. Here are the top 20.
We found 82 Reddit comments discussing the best graph theory books. We ranked the 31 resulting products by number of redditors who mentioned them. Here are the top 20.
Here's my rough list of textbook recommendations. There are a ton of Dover paperbacks that I didn't put on here, since they're not as widely used, but they are really great and really cheap.
Amazon search for Dover Books on mathematics
There's also this great list of undergraduate books in math that has become sort of famous: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm
Pre-Calculus / Problem-Solving
Calculus
Linear Algebra
Differential Equations
Number Theory
Proof-Writing
Analysis
Complex Analysis
Functional Analysis
Partial Differential Equations
Higher-dimensional Calculus and Differential Geometry
Abstract Algebra
Geometry
Topology
Set Theory and Logic
Combinatorics / Discrete Math
Graph Theory
P. S., if you Google search any of the topics above, you are likely to find many resources. You can find a lot of lecture notes by searching, say, "real analysis lecture notes filetype:pdf site:.edu"
You are in a very special position right now where many interesing fields of mathematics are suddenly accessible to you. There are many directions you could head. If your experience is limited to calculus, some of these may look very strange indeed, and perhaps that is enticing. That was certainly the case for me.
Here are a few subject areas in which you may be interested. I'll link you to Dover books on the topics, which are always cheap and generally good.
Basically, don't limit yourself to the track you see before you. Explore and enjoy.
In the case of this paper, it's referring to dimensions in a mathematical sense, not a physical "space-like" or "time-like" sense. In that regard, the more abstract mathematical notion of "dimension" is used all the time to describe things on a computational level that most people wouldn't associate with their idea of "dimension". For example, a picture on the computer can be thought of as a single point in some extremely high dimensional space (Im talking on the scale of millions of dimensions).
Personally, I'd find a more interesting occult correlation between the neural network structure shapes being directed/undirect simplices. If anyone is curious about learning about some of the mathematics behind those sorts of structures (called graphs) I'd recommend Introduction to Graph Theory by Dover books on the subject. It's a great introduction and has a great preface on the subject of mathematics.
The GTM Graph Theory book is ultra dense and wonderful. http://www.amazon.com/Graph-Theory-Graduate-Texts-Mathematics/dp/3540261834
Algebra
Trigonometry
Functions and Graphs
These are three books that I would recommend to somebody trying to prepare for calculus. They're all written by the mathematician Gelfand and his colleages, and they're some of the best-written math books I've ever read. You come away from reading them really understanding the subject matter. I'd read them in that order, too.
Here is the book I always recommend for people who want an introduction to graph theory:
It's super cheap (only $3.99 on Amazon) and I think it's really a good introduction to the subject. It doesn't go as far in depth as more advanced books, but Kuratowski's theorem is covered in Chapter 3.
Anything by Bollobas is great (graduate student level). Frank Haray's work is very accessible. Also, Combinatorics and Graph Theory is a brilliant introductory text for undergraduates or newcomers to the field.
Read the "Introductory Graph Theory" by G. Chartrand. It's pretty accessible, the theorems' proofs are well written and it has lots of interesting problems to do on your own (with some hints/solutions at the end). I found it great for self-study, especially for someone without any experience with proofs, like me. Enjoy !
A First Course in Graph Theory by Chartrand and Zhang
Combinatorics: A Guided Tour by Mazur
Discrete Math by Epp
For Linear Algebra I like these below:
Lecture Notes by Tao
Linear Algebra: An Introduction to Abstract Mathematics by Robert Valenza
Linear Algebra Done Right by Axler
Linear Algebra by Friedberg, Insel and Spence
There are some really good books that you can use to give yourself a solid foundation for further self-study in mathematics. I've used them myself. The great thing about this type of book is that you can just do the exercises from one side of the book to the other and then be confident in the knowledge that you understand the material. It's nice! Here are my recommendations:
First off, three books on the basics of algebra, trigonometry, and functions and graphs. They're all by a guy called Israel Gelfand, and they're good: Algebra, Trigonometry, and Functions and Graphs.
Next, one of two books (they occupy the same niche, material-wise) on general proof and problem-solving methods. These get you in the headspace of constructing proofs, which is really good. As someone with a bachelors in math, it's disheartening to see that proofs are misunderstood and often disliked by students. The whole point of learning and understanding proofs (and reproducing them yourself) is so that you gain an understanding of the why of the problem under consideration, not just the how... Anyways, I'm rambling! Here they are: How To Prove It: A Structured Approach and How To Solve It.
And finally a book which is a little bit more terse than the others, but which serves to reinforce the key concepts: Basic Mathematics.
After that you have the basics needed to take on any math textbook you like really - beginning from the foundational subjects and working your way upwards, of course. For example, if you wanted to improve your linear algebra skills (e.g. suppose you wanted to learn a bit of machine learning) you could just study a textbook like Linear Algebra Done Right.
The hard part about this method is that it takes a lot of practice to get used to learning from a book. But that's also the upside of it because whenever you're studying it, you're really studying it. It's a pretty straightforward process (bar the moments of frustration, of course).
If you have any other questions about learning math, shoot me a PM. :)
Many thanks for the suggestions!
For the interested, I bought this book for GT:
http://www.amazon.com/Introductory-Graph-Theory-Gary-Chartrand/dp/0486247759
I also was tempted by the following book:
http://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewart/dp/0486284247
I think buying a book feels better than sex. (I can compare.)
Ah, yes! Miklos Bona wrote a book titled A Walk Through Combinatorics, which is by far my favorite "new" book in a long time. It deals with the very basics, and moves through some pretty complex theory with amazingly fun and insightful problems.
This book is the current first course combinatorics text at MIT. Bona is a professor at UF now.
Neat - there's someone is interested in the same field as me. Here's some resources on the topic.
Dr. Fan Chung of UCSD is considered one of the premier experts in the field. She has plenty of resources on it on her website. Link to her website
Dr. Steve Butler from Iowa State (who studied under Dr. Chung) has a playlist of his course lectures on it. Each video has a link to the notes. Link to playlist
Another expert on the subject I've met is Dr. Paul Horn, another student of Dr. Chung. Link to his website
Books:
Godsil and Royle's book on Algebraic Graph Theory. This is a good intro to the combined study of algebra and graph theory and has a couple chapters all about graph matrices. I would consider it essential for someone just getting into the topic, especially as an upper-level undergraduate.
Svetkovic, Rowlinson, & Simic's "An Introduction to the Theory of Graph Spectra". I consider this to be a little light on depth in any topic but it gives you an exceptionally diverse taste of all the different types of problems in the field.
Dr. Chung's lecture notes on spectral graph theory. She compiled these notes from a series of talks she did on the subject in the 90's. The notes have more depth but less breadth than the previous book I listed. Also not to brag or anything, but I got my copy signed by her at last year's JMM.
I hope these resources will be useful to you. Happy hunting.
You could start by going through here and seeing if anything catches your eye but if your still in high school I suppose you might not know what to look for. When I was in high school (currently an undergraduate in math) this book was one that really made me first consider the idea of trying to become a mathematician: R. Trudeau's Introduction to Graph Theory. It is a pretty short read but gives a very nice introduction to graph theory and what pure mathematics is all about.
They may not be the best books for complete self-learning, but I have a whole bookshelf of the small introductory topic books published by Dover- books like An Introduction to Graph Theory, Number Theory, An Introduction to Information Theory, etc. The book are very cheap, usually $4-$14. The books are written in various ways, for instance the Number Theory book is highly proof and problem based if I remember correctly... whereas the Information Theory book is more of a straightforward natural-language summary of work by Claude Shannon et al. I still find them all great value and great to blast through in a weekend to brush up to a new topic. I'd pair each one with a real learning text with problem sets etc, and read the Dover book first quickly which introduces the reader to any unfamiliar terminology that may be needed before jumping into other step by step learning texts.
Check out this list. It includes most if not all books mentioned here.
From more to less advanced, generally:
A Mathematical Gift, II
A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra (Mathematical World) (v. 3)
Note the first three volumes are cheaper as a set.
Intuitive Topology (Mathematical World, Vol 4)
Groups and Symmetry: A Guide to Discovering Mathematics (Mathematical World, Vol. 5)
Knots and Surfaces: A Guide to Discovering Mathematics (Mathematical World, Vol. 6)
and 5 activity books in the same style
He really should be starting with the Trudeau, much better bed side reading.
For proofs in general, I like D'Angelo and West's Mathematical Thinking. http://www.amazon.com/Mathematical-Thinking-Problem-Solving-Proofs-Edition/dp/0130144126
For discrete math, especially combinatorics, I loved Miklos Bona's A Walk Through Combinatorics. http://www.amazon.com/Walk-Through-Combinatorics-Introduction-Enumeration/dp/9814335231/
For induction proofs, you check your base case, assume the induction hypothesis (true for k), and then check k+1.
You should be able to manipulate the k+1 term into something involving the k term, and that will then lead to the k+1 conclusion.
Example For all n >= 4, 2^(n) < n!
Base case: n = 4. 2^(4) = 16 < 24 = 4!
IH: Assume true for some k >= 4.
Then 2^(k+1) = 2*2^(k)
2*2^(k) < 2*k! (Induction Hypothesis used here)
2*k! < (k+1)k! (k > 3, so k+1 > 2)
(k+1)k! = (k+1)! (definition of factorial)
Yea John Green certainly isn't for everyone, particularly outside of the YA target audience. I wouldn't say it's his strongest book either, but it might be useful to check out.
In terms of mathematical directions you could go, graph theory is actually a pretty solid field to work in. It's basics are easy to grasp, the open problems are easy to understand and explain, and there are many obscure open ones that are easily within reach of a talented high schooler. In fact a lot of combinatorics is like that as well. I would recommend the book Introduction to Graph theory by Trudeau (which was originally titled Dot's and Lines). It's a great introduction to mathematical proof while leading the reader to the forefront of graph theory.
Is this the book you are looking for https://www.amazon.com/Functions-Graphs-Dover-Books-Mathematics/dp/0486425649/ref=cm_cr_arp_d_pdt_img_top?ie=UTF8 ?
There are more freely available books from erstwhile USSR published by Mir Publishers https://mirtitles.org/
From the ground up, I dunno. But I looked through my amazon order history for the past 10 years and I can say that I personally enjoyed reading the following math books:
An Introduction to Graph Theory
Introduction to Topology
Coding the Matrix: Linear Algebra through Applications to Computer Science
A Book of Abstract Algebra
An Introduction to Information Theory
To pedal off of this, graph theory is pretty much everywhere and it's really straightforward to learn. This is a really good intro book and it's really cheap.
These are, in my opinion, some of the best books for learning high school level math:
These are all 1900's Russian math text books (probably the type that /u/oneorangehat was thinking of) edited by I.M. Galfand, who was something like the head of the Russian School for Correspondence. I basically lived off them during my first years of high school. They are pretty much exactly what you said you wanted; they have no pictures (except for graphs and diagrams), no useless information, and lots of great problems and explanations :) There is also I.M Gelfand Trigonometry {[.pdf] (http://users.auth.gr/~siskakis/GelfandSaul-Trigonometry.pdf) | Amazon} (which may be what you mean when you say precal, I'm not sure), but I do not own this myself and thus cannot say if it is as good as the others :)
I should mention that these books start off with problems and ideas that are pretty easy, but quickly become increasingly complicated as you progress. There are also a lot of problems that require very little actual math knowledge, but a lot of ingenuity.
Sorry for bad Englando, It is my native language but I haven't had time to learn it yet.
I'd recommend doing mathematics, It's much important than learning a language. It helps you grab the logic of solving a problem.
Discrete Mathematics by Rosen is the best book from my experience.
Graph Theory by Bollobas is recommended by many but i prefer Graph Theory by Douglas West
Algorithms by Cormen. No introductions needed this book encompasses most of the problems you'll encounter.
However if you're keen on learning a C/C++/Java i'd recommend the Head First Series from O'Reily .
Goodluck!
It makes me so happy to see this post! I can't believe that someone loves graph theory as much as I do, I wish my roommate was cool enough to get me a new graph theory text :)
One of my favorites is
http://www.amazon.com/Algebraic-Graph-Theory-Chris-Godsil/dp/0387952209
It's a text on algebraic graph theory, a bit more specialized than a lot of the books suggested here. If your friend doesn't have any books on algebraic graph theory then this should offer him quite a bit of new material to read over!
If you need to brush up on some of the more basic topics, there's a series of books by IM Gelfand:
Algebra
Trigonometry
Functions and Graphs
The Method of Coordinates
I can see how that could be an issue.
Some books on amazon have a few pages out of the middle in the preview, and sometimes if you hit 'surprise me' you can get some pages not available otherwise.
If worse comes to worst, just ask here!
By the way, if you want a graph theory text without any 'fluff' at the beginning, look at Diestel. It's labeled as a grad text, and it does move rather quickly, but my undergrad graph theory course used this text, and I really like it.
Read this: https://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773 and you're more than set for algebraic manipulation.
And if you're looking to get super fancy, then some of that: https://www.amazon.com/Method-Coordinates-Dover-Books-Mathematics/dp/0486425657/
And some of this for graphing practice: https://www.amazon.com/Functions-Graphs-Dover-Books-Mathematics/dp/0486425649/
And if you're looking to be a sage, these: https://www.amazon.com/Kiselevs-Geometry-Book-I-Planimetry/dp/0977985202/ + https://www.amazon.com/Kiselevs-Geometry-Book-II-Stereometry/dp/0977985210/
If you're uncomfortable with mental manipulation of geometric objects, then, before anything else, have a crack at this: https://www.amazon.com/Introduction-Graph-Theory-Dover-Mathematics/dp/0486678709/
It's my understanding that UC Berkeley uses the following text:
http://www.amazon.com/Walk-Through-Combinatorics-Miklos-Bona/dp/9810249012
Might be worth looking into!
Combinatorics and Graph Theory by Harris et al is a nice and short introduction to combinatorics, and even includes some interesting/exotic material not usually found in the introductory combinatorics books. Grimaldi is more comprehensive of course, but also more expensive.
As I see it there are four kinds of books that fall into the sub $30 zone:
You can get a lot of great books if you are willing to spend a bit more however. For example, Hardy and Wright is an excellent book (and if you think about it: is a 600 page book for $60 really more expensive than a 300 page one for 30?). Richard Stanley's books on combinatorics: Enumerative Combinatorics Vol. I and Algebraic Combinatorics are also excellent choices. For algebra, Commutative Algebra by Eisenbud is great. If computer science interests you you can find commutative algebra books with an emphasis on Gröbner bases or on algorithmic number theory.
So that's a lot of suggestions, but two of them are free so you can't go wrong with those.
Even low-end soroban suit my needs (and I'm pretty picky about having good tools). I think I have a few that are around this quality: https://www.amazon.com/Flexzion-Arithmetic-Calculating-Oriental-Calculator/dp/B018SA829E
The only bad experience I've had is with some of the wooden abacuses being low quality and splintering on the back. For a younger child, I might just want to go plastic.
To teach younger kids, I recommend the SAI Speed Academy's series: https://www.amazon.com/Abacus-Mind-Math-Instruction-Level/dp/1941589006
I don't know about other bases and the soroban, though. Sorry!
What are you majoring in?
What you're describing could just be a personality issue that's unrelated to maths, that maths is just be an example of. That being said, I find the way people are taught maths to be a form of abuse. It's like the way someone who was molested as a child might have weird issues with sex, so do most people have issues with maths who have had to go through maths in high school.
Just so that you know, what you think maths is, is actually almost not at all what maths really is. I would recommend, after you finish your exams and have nothing better to do, read this book about graph theory. It's $4 + shipping from amazon, or you may have it in the library wherever you're studying. It's kind of pointless, but there are a few nice bits about the philosophy of maths.
Since last we spoke, I have mostly been reading:
Today I purchased:
Specifically Graph Theory? Pearls In Graph Theory was the one that I used in my undergraduate course. It covers some algorithms and the theory behind completeness, infinite lattices, etc. Fun stuff!
If you want a more combinatorial view of things (ie: not so specialized with regards to Graph Theory), I highly recommend Brualdi's Introductory Conbinatorics as bucket1004 said. I also used that one for my combinatorics course and I loved it.
Disclaimer: There's some errata in the edition that I used. I'm not so sure how much better the new editions are.
Here are some great books that I believe you may find helpful :)
and last but definitely not least:
Later on:
Chapters are short and engaging. Great exercises. Very visual and creative which should make mathematical structure more appealing to a lay person.
I would recommend these, as well as
http://www.math.upenn.edu/~wilf/DownldGF.html
http://www.amazon.com/Walk-Through-Combinatorics-Introduction-Enumeration/dp/9814335231
http://www.amazon.com/Introduction-Graph-Theory-Douglas-West/dp/0130144002/
I think you might like this.
Introduction to Graph Theory - Chartrand
It tries to motivate most all the topics by either
problems, puzzles, or history.
I've already taken linear algebra, I had posted on here at a different time and got the feeling my class wasn't quite complete so I got a book that covered everything we missed.
The Graph Theory book is A First Look at Graph Theory.
I'm not too worried about not being at the right level for these books, I'm more worried that because there's no class and no lectures I won't remember anything without taking notes. But I suck at taking notes from books.
D.B. West is an extremely challenging book for self study due to its terseness, much like many typical graduate level courses. For a first course in graph theory, I highly recommend "A First Course in Graph Theory" by Chartrand and Zhang (appropriately named). The exposition is lucid and beautiful, and kept to a reasonable amount. The material is also motivated, and was overall a great experience for me. This was one of the first math books I read, and I think it is quite amazing. I did have to use D.B. West's book in a later graph theory course, so I do know what using that one is like: let's just say, not fun.
https://www.amazon.com/First-Course-Graph-Theory-Mathematics/dp/0486483681/ref=tmm_pap_swatch_0?_encoding=UTF8&amp;qid=&amp;sr=
Sure, there are a few directions you could go:
Algorithms: A basic understanding of how to think about and analyze algorithms is pretty necessary if you were to go into combinatorial optimization and is a generally useful topic to know in general. CLRS is the most famous introductory book on algorithms, and it gets the job done. It's long, but I thought it was decent enough. There are also plenty of video lectures on algorithms online; I liked the MIT OpenCourseWare of this class.
Graph Theory: Many combinatorial optimization problems involve graphs, so you would definitely want to know some graph theory. It's also super interesting, and definitely worth learning regardless! West is a good book with lots of exercises. Bondy and Murty and Diestel also have good books, which are freely available in PDF if you do a google search. Since you're doing a project on traffic optimization, you might find network flows interesting. Networks are directed graphs, where you think about moving "flow" across the edges of the graph, so they are useful for modelling a lot of real-life problems, including traffic. Ahuja is the best book I know on network flows.
Linear and Integer Programming: Many optimization problems can be described as maximizing (or minimizing) some linear function subject to a set of linear constraints. These are linear programs (LPs). If the variables need to take on integer values, then you have an integer program (IP). Most combinatorial optimization problems can be formulated as integer programs. Integer programming is NP-hard, but in practice there are methods that can solve most IPs , even very large ones, relatively quickly. So, if you actually want to optimize things in real-life this is a very useful thing to know. There's also a mathematically rich field of developing methods to solve IPs. It's a bit of a different flavor than the rest of this stuff, but it's definitely a fertile area of research. Bertsimas is good for learning linear programming. Unfortunately, I don't have a good recommendation for learning integer programming from scratch. Perhaps the chapters in Papadimitriou - Combinatorial Optimization would be a good introduction.
Approximation Algorithms: This is about algorithms which quickly (in polynomial time) find provably good but not necessarily optimal solutions to NP-hard problems. Williamson and Shmoys have a great book that is freely available here.
The last book I'd recommend is Schrijver. This is the bible for the field. I put it here at the end because it's more of a reference book rather than something you could read cover to cover, but it's REALLY good.
Lastly, if you like traffic optimization, maybe look up what people are doing in operations research departments. A lot of OR is about modelling real problems with math and analyzing the models, so this would include things like traffic optimization, vehicle routing problems, designing smart electric grids, financial engineering, etc.
Edit: Not sure why my links aren't all formatting correctly... sorry!
Here's the one I used for the class I took. It's a bit more of a reference though
https://www.amazon.com/Introduction-Graph-Theory-Douglas-West/dp/0130144002/ref=sr_1_18?ie=UTF8&amp;qid=1510241743&amp;sr=8-18&amp;keywords=graph+theory+textbook
I know this is removed, so I can recommend my tool which builds a graph of products that are often bought together at Amazon.
http://www.yasiv.com/#/Search?q=graph%20theory&amp;category=Books&amp;lang=US - this is a network of books related to graph theory. Finding the most connected product usually yields a good recommendation. In this case it recommends to take a deeper look at https://www.amazon.com/Introduction-Graph-Theory-Dover-Mathematics/dp/0486678709
This technique involves assuming a constant circumsphere radius between each solid, I.E. they each inscribe the same sphere with their vertices. This chapter demonstrates how this assumption can be used to draw the faces of each using radial lines of 6, 8 and 10 fold symmetry (drawing these is covered in earlier chapters). I am also sharing the first two chapters of this book here.
https://drive.google.com/open?id=0B75RWn1eZrL0Y0FLdHJVYmdIR3c
If you would like the full version, a paperback version can be purchased on Amazon here.
https://www.amazon.com/Key-Platonic-Solids-Straightedge-Construction/dp/1544685017/ref=sr_1_1?ie=UTF8&amp;qid=1519617508&amp;sr=8-1&amp;keywords=key+to+the+platonic+solids
Or you can purchase a digital PDF copy for download here.
https://cyborgclothes.com/products/key-to-the-platonic-solids-e-book
As stated in the title. These methods can be used to draw each of the orthogonal projections, including the face, edge, and vertex in both the centered and normal position. Thanks for stopping in and please feel free to ask anything you like.
I like reading math books for fun, especially cheap Dover books. Excursions in Number Theory by Ogilvy & Anderson (lots of cool little stuff). Introductory Graph Theory by Chartrand (a lot of real-world programming boils down to graph theory). An Introduction to Algebraic Structures by Landin (abstract algebra). In Code: A Mathematical Journey by Flannery (modular arithmetic, factoring, and cryptography). In Code or Excursions would probably help with Project Euler 3 and several others.
If anyone is interested in learning more about graph theory, this is a great (and brief) book that requires very little mathematical background. I highly recommend it.
http://www.amazon.com/Introductory-Graph-Theory-Gary-Chartrand/dp/0486247759
Welcome to graph theory!
I liked the Harris, Hirst, Mossinghoff text that I used for an undergrad class. It's a solid intro that builds from the very beginning, and at times has an almost irreverent writing style that references everything from Shakespeare to Kevin Bacon.
>I like how you came here to make a distinction without a difference
That you think these sets are equivalent is the problem with "STEM" in this country. I'm not blaming you, it's not your fault. For whatever reason, set theory is barely discussed. Even in multivariate calculus, the most you care about sets is with domain and range, just like in algebra. Here are a few topics that are mathematics, and not arithmetic:
-Set Theory
-Topology (Better than Munkres)
-Graph Theory
-Abstract Algebra (Groups/Rings/Fields)
Basic quantifiers pop up first in set theory, which as far as I can tell is only recommended after integral calculus. Things like ∀, and ∃ have a particular meaning, and their orders and quantities are very specific.
If you would like to know more about the difference between mathematics and arithmetic (which is a subset), then start with set theory. You'll need that to do anything else. I can try to answer any other questions you may have.
My go to book for anything graph theory related is the intro book by West.
Great book for undergrad / first year grad students. Goes into detail on numerous topics and if I can recall, you can find a bit of good application there. I know computer science replies on applications of graph theory quite a bit, so you may be able to delve further into that.
Not a book, but you might like this: https://dev.to/vaidehijoshi/a-gentle-introduction-to-graph-theory
As for a book, try this: https://www.amazon.com/Introduction-Graph-Theory-Dover-Mathematics/dp/0486678709
For compsci you need to study tons and tons and tons of discrete math. That means you don't need much of analysis business(too continuous). Instead you want to study combinatorics, graph theory, number theory, abstract algebra and the like.
Intro to math language(several of several million existing books on the topic). You want to study several books because what's overlooked by one author will be covered by another:
Discrete Mathematics with Applications by Susanna Epp
Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand, Albert D. Polimeni, Ping Zhang
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
Numbers and Proofs by Allenby
Mathematics: A Discrete Introduction by Edward Scheinerman
How to Prove It: A Structured Approach by Daniel Velleman
Theorems, Corollaries, Lemmas, and Methods of Proof by Richard Rossi
Some special topics(elementary treatment):
Rings, Fields and Groups: An Introduction to Abstract Algebra by R. B. J. T. Allenby
A Friendly Introduction to Number Theory Joseph Silverman
Elements of Number Theory by John Stillwell
A Primer in Combinatorics by Kheyfits
Counting by Khee Meng Koh
Combinatorics: A Guided Tour by David Mazur
Just a nice bunch of related books great to have read:
generatingfunctionology by Herbert Wilf
The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates by by Manuel Kauers, Peter Paule
A = B by Marko Petkovsek, Herbert S Wilf, Doron Zeilberger
If you wanna do graphics stuff, you wanna do some applied Linear Algebra:
Linear Algebra by Allenby
Linear Algebra Through Geometry by Thomas Banchoff, John Wermer
Linear Algebra by Richard Bronson, Gabriel B. Costa, John T. Saccoman
Best of Luck.
I haven't found any books I really care for, but Dan Spielman has some pretty good notes from a course here. Fan Chung has a book which is partially available online. Another term to look for is algebraic graph theory. There's a reasonably good book with that title.
First, please make sure everyone understands they are capable of teaching the entire subject without a textbook. "What am I to teach?" is answered by the Common Core standards. I think it's best to free teachers from the tyranny of textbooks and the entire educational system from the tyranny of textbook publishers. If teachers never address this, it'll likely never change.
Here are a few I think are capable to being used but are not part of a larger series to adopt beyond one course:
Most any book by Serge Lang, books written by mathematicians and without a host of co-writers and editors are more interesting, cover the same topics, more in depth, less bells, whistles, fluff, and unneeded pictures and other distracting things, and most of all, tell a coherent story and argument:
Geometry and solutions
Basic Mathematics is a precalculus book, but might work with some supplementary work for other classes.
A First Course in Calculus
For advanced students, and possibly just a good teacher with all students, the Art of Problem Solving series are very good books:
Middle & high school:
and elementary linked from their main page. I have seen the latter myself.
Some more very good books that should be used more, by Gelfand:
The Method of Coordinates
Functions and Graphs
Algebra
Trigonometry
Lines and Curves: A Practical Geometry Handbook