I am an undergraduate who had recently finished his $2$nd year. I was wondering how far can I get with Graph Theory this summer.
I am studying from Bondy & Murty's book. I already finished Chapter $1$, along with its exercises, and I am getting a hang of its basic concepts.
My background with algebra is: a lot of linear algebra, and a good grasp of basic and some advanced group theory as well as ring theory.
I am an average student (i.e., with not much extra genetic powers).
How much could I get far with graph theory in two months, if I dedicate about $2$-$3$ hours for it per day?
Thank you.
The answer to this question is: "How would anyone know it?" - Unless there are some people on MSE that do have a very sensitive extrasensory perception and could predict what happens with you$^{[1]}$ - A better question you could make$^{[2]}$ is "In a course of graph theory, what are the students expected to learn after $2$ months studying $2-3$ hours a day?"
For that answer, you can take a syllabus of a course in graph theory and divide the contents by $3$. For example, you can take this one. I recommend you to not just read, try to watch some lectures, for example these lectures. Some more advanced proofs in graph theory demand that you see someone doing step-by-step, but that depends on the way you learn things, some people could find it easier to just read instead of watching lectures.
$[1]:$ Their existence would invalidate part of my answer.
$[2]:$ Obviously, assuming we don't have users with ESP.