Research on ground state configuration of Ising model I want to do mathematical research (algorithm construction and mathematical analysis) on Ising model ground state configuration. From what I know, the state of art research is using graph theory formalism. 
Could someone give me some advice on how to learn this subject (say, what books or papers to read) to a research level from a basic undergraduate math background?
 A: This is interesting.  I actually wrote some code about 6 months ago which applied the Ising model to networks (represented as graphs).  You can check out the code on github here if you like.  It is written in Python and uses a few awesome libraries which make working with graphs a breeze.  It is in the development stages, as I did not have time to proceed much beyond getting it to work and look for some phase transitions in certain classes of networks.
As per your questions:  I learned graph theory in a graduate mathematics course from the text by Merris called 'Graph Theory'.  It is a great place to start, as it is quite accessible to undergraduates and really doesn't require that you know much beyond the concepts and usage of the material presented in an introductory abstract algebra course (groups, rings, etc..).  There is a good text by Marshall called Applied Graph Theory.  I believe that your research may lead you to studying the eigenspaces of graphs, in which case I recommend you look for the text by Cvetkovic, Rowlinson, Simic called Eigenspaces of Graphs (Encyclopedia of Mathematics).
