n this talk we show how graph theory provides a unifying framework to model, analyze, and solve complex optimization problems, as those arising in stochastic optimization (graph is a tree), optimal control and state estimation (graph is a line), PDE-constrained optimization (graph is a mesh), network optimization (graph is the network), and combinations (e.g., stochastic PDE-constrained problems). We show how the graph abstraction facilitates the derivation of theoretical properties of optimization problems. Specifically, we show that, under standard regularity conditions, nonconvex continuous optimization problems posses a property that we call exponential decay of sensitivity (EDS); this property indicates that the effect of parametric perturbations decays exponentially along the graph. We show that how EDS relates to controllability and observability properties of optimal control problems. In addition, we show that EDS can help derive graph coarsening and decomposition strategies to obtain approximate solutions of high quality. We also show how the graph abstraction facilitates the derivation of new algorithmic strategies to solve large-scale problems in distributed settings. Specifically, we introduce a new decomposition paradigm that we call overlapping Schwarz decomposition, which facilitates the implementation of distributions optimization settings with various degrees of communication. Finally, we discuss how graph theory can be used to model digital twins and to understand how data and decisions propagate through the structure of complex systems. This is work in collaboration with Sungho Shin (MIT), Sen Na (Georgia Tech), and Mihai Anitescu (Argonne)