In the last years different techniques coming from algebraic geometry have been used also in different fields of mathematics. For example, the concept of divisor, that gave new insight in the study of algebraic curves, has been introduced in the theory of graphs leading to important results. In fact, analogously as on algebraic curves, linear systems are introduced on graphs/metric graphs/tropical curves. It is shown that linear systems on these combinatorial objects also satisfy e.g. the Riemann-Roch theorem, the Abel-Jacobi theory and the Clifford inequality. On the other hand, using Baker's specialization lemma, they can be used to study algebraic curves. The main subject of this thesis is the specialization of linear systems from curves to graphs. This is a very recent topic of research that combines combinatorial, tropical, non-Archimedean and algebraic geometry.

The first three chapters have a strong combinatorial flavor. After a brief introduction on linear systems on (metric) graphs and on harmonic morphisms in Chapter 1, we focus on complete graphs. In Chapter 2 we compute the gonality sequence of complete graphs and of the corresponding metric graphs. This turns out to be a quite interesting combinatorial problem. Then, we consider complete metric graphs with arbitrary edge lengths. We compute the gonality of complete graphs with a small number of omitted edges. In Chapter 3 we consider similar results for bipartite graphs.

The following two chapters are the geometrical core of the thesis. In Chapter 4 we explain how to specialize divisors from curves to metric graphs, emphasizing how Berkovich's non-Archimedean geometry presents the right setting to do so. We present a first example on how to use the specialization map and the gonality sequence of smooth plane curves to deduce an upper bound on the gonality sequence of complete graphs. Then in Chapter 5, we check whether for the graphs and the linear systems considered in Chapter 2 there exist curves and linear systems of the same rank specializing to them. These types of problems are called lifting problems. In particular, we lift the graphs using models of plane curves with certain singularities. We conclude by using lifts of harmonic morphisms and the specialization map to study the specialization of theta characteristics on hyperelliptic curves and graphs.