One could likewise argue that General Algebra (which encompasses groups, rings, lattices, and many other structures) is the study of a special class of categories, and so argue that General Algebra is a branch of Category Theory... (For example, George Bergman's General Algebra book is heavily category-theoretically flavored).
I would say rather that Category Theory has some very large areas of intersection with General/Abstract Algebra; I remember George Bergman saying once that in the 80s (?) there was a big conference at Berkeley/MSRI that invited both Universal Algebraists and Category Theorists, and that many times over the course of the conference they discovered that there were results that each "camp" had proven independently and did not realize the other "side" knew about them, or that there were questions that had been raised on the periphery of one whose answer was well-known by the other. (I hope I'm not misremembering and/or misreporting this!).
My particular (heavily algebra-biased) experience leads me to think that Category Theory is closer to abstract algebra than to other branches of pure mathematics (e.g., topology, analysis, etc). Don't know if I would go so far as to call it a "branch" of abstract algebra, though, any more than I would call it a branch of set theory (or set theory a branch of category theory).
The book categories for the working mathematician, by Saunder's MacLane, comes well-recommended. Although I have never quite got around to reading it, it has such a promising title! Note that Saunder's MacLane was one of the original category theorists.
For an explicit example of graph theory and category theory working together to prove results on groups, there is a well-known link between the category of graphs and subgroups of free groups. The classical reference is Topology of Finite Graphs by Stallings. For example, if the non-diagonal component of the fibre product of a graph with itself is simply connected then the corresponding subgroup is malnormal. Dani Wise lifted this idea to the more general category of "cubical complexes", which allowed him to prove some famous open problems in group theory and G&T (for example, he proved the virtually Haken conjecture, and that every one-relator group with torsion is residually finite). I found Wise's paper The residual finiteness of positive one-relator groups to be especially helpful.
You say you want to use category theory and graphs to look at endomorphisms of groups. Well, one of the questions I was attacking in my PhD thesis was the following.
Fix a class of groups $\mathcal{C}$. Does every (countable) group occur as the outer automorphism group of a group from the class $\mathcal{C}$?
I fixed a class $\mathcal{C}$, and I managed to prove that the above result held for this class so long as I could prove that a certain group (in reality, a class of very similar groups) had a malnormal subgroup (with certain additional properties). So, I then took the fibre product of a "subgroup" in the ambient free group and (with a bit of effort) proved that its malnormality fell down to my group.
This doesn't quite answer your question, but I thought it relevant enough to mention. If you want, I can send you a copy of my thesis.
Best Answer
Universal algebra discusses algebraic systems such as groups,rings,etc. independent of elements or specific examples of such systems-it discusses algebraic systems in general in terms of the operators and relations between those elements only. A algebraic system is defined as a nonempty set with at least one n-ary operation on it. We discuss then a specific kind of algebraic system and it's operations.For example, in universal algebra, we discuss the collection of all groups as a set with an binary associative operation and 2 UNARY operations corresponding to the general identity and the inverse of each element. No specifics about the elements are allowed to be discussed, only general principles unique to groups. Equational relations are added as axioms. In short, it is strictly a "big picture" approach to algebra.But note it's different from the "big picture" approach of category theory since it only discusses one kind of object at a time and does not consider the relations between collections of different kinds of objects.
Category theory takes this one step further by discussing the operations and relations between different kinds of collections of objects-note the objects do not necessarily have to be algebraic systems- codified by functors and commutative diagrams.
In many ways. category theory can be seen as a direct generalization of universal algebra the same way point set topology can be seen as a generalization of ordinary calculus,real and complex analysis. As point set topology strips away the specific algebraic and ordering properties of the real and complex Euclidean spaces to lay bare the common structures that makes continuity and convergence possible on such systems, category theory allows one to discuss the relations between collections of "the same" objects while universal algebra discusses the internal operations of single categories of a single kind-namely, algebraic systems.
At least,that's how I understand it. That help?