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 isnt 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.