There is an interaction between category theory and graph theory in
F.~W.~Lawvere. Qualitative distinctions between some toposes of generalized graphs. In {\em Categories in computer science and logic (Boulder, CO, 1987)/}, volume~92 of {\em Contemp. Math./}, 261--299. Amer. Math. Soc., Providence, RI (1989).
which we have exploited in
R. Brown, I. Morris, J. Shrimpton and C.D. Wensley, `Graphs of Morphisms of Graphs', Electronic Journal of Combinatorics, A1 of Volume 15(1), 2008. 1-28.
But that is actually about possible categories of graphs, which may be the opposite of the question you ask.
If you look at groupoid theory, then "underlying graphs" are fundamental, for example in defining free groupoids. See for example
Higgins, P.~J. Notes on categories and groupoids, Mathematical Studies, Volume~32. Van Nostrand Reinhold Co. London (1971); Reprints in Theory and Applications of Categories, No. 7 (2005) pp 1-195.
Groupoids are kind of "group theory + graphs".
The fundamental result the completely characterizes recurrent/transient graphs is that a graph is recurrent if and only if the effective resistance of the graph, when considered as an electric network where every edge has resistance one, from some/any vertex to infinity is infinite. This is true also when the degrees are unbounded.
Of course, to use this, you need to understand how this resistance is defined and what method there are to show it's finite/infinite. The easiest ways to bound the the resistance from below is finding cutsets, sets which separate some specified vertex from infinity. If you have a disjoint sequence of such cutsets of sizes $a_n$, the effective resistance is at least $\sum_n 1/a_n$, so if this sum is infinite, the graph is recurrent. This is called the Nash-Williams criteria and it's easy to see that graphs growing slower than quadratic satisfy it (in fact, a slightly bigger growth rate is still OK).
In the other direction, one can bound the resistance from above by using flows. This is slightly more involved, I can elaborate later, if there is a demand.
Best Answer
A self study course I can recommend for topology is Topology by JR Munkres followed by Algebraic Topology by A Hatcher (freely and legally available online, courtesy of the author!). But that is if you want to be able to really do the math in all its glorious detail. Basic Topology by MA Armstrong is a shortcut and a very good one at that.
The closest I can get to what you are asking for here is Network Topology. Is that what you mean? In that case you should be probably be looking at topological graph theory. Wikipedia also tells me that something called Computational Topology exists, but that is probably not what you are looking for.
Hope that helps!