Jürgen Elstrodt - Maß- und Integrationstheorie (only in German)
Fremlin - Measure Theory (freely available in the web space, contains pretty much every significant aspect of measure theory in appropriate depth)
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".
Best Answer
Diestel's book is not exactly light reading but it's thorough, current and really good. Also in the GTM series is Bollobas' book which is very good as well, and covers somewhat different ground with a different angle (Diestel emphasizes the forcing relationships between various invariants which is a nice unifying theme).
There are hundreds of other introductory texts, but I would go with one of these two (or both).