1. The theory of continuous graphs also known as graphons heavily relies on functional analysis of the Lebesgue space $L_1$ as one has to deal with certain limits of discrete objects with respect to the norm of $L_1$.
2. There is a characterisation of super-reflexive Banach space via impossibility of metric embedding of certain graphs (diamond graphs) into Banach spaces. See papers by W.B. Johnson, A. Naor, M. Ostrovskii or G. Schechtman. For example: W. B. Johnson, G. Schechtman, Diamond graphs and super-reflexivity, J. Topol. Anal., 1 (2009), 2, 177–189.
3. There is a huge subfield of the theory operator algebras devoted to encoding graphs by certain C*-algebras (the so-called graph algebras). For example they generalise the Cuntz algebras which in a sense are building blocks of purely infinite algebras that are classifiable in certain sense. See these introductory notes.

---

Source: https://math.stackexchange.com/questions/1664403/links-between-functional-analysis-and-graph-theory