The aim of this project consists in developing and extending the relationship among different mathematical fields: group theory, combinatorics, graph theory and computer science. Group theory has found, by the introduction of self-similar groups acting on rooted trees, a new class of examples with special and exotic properties which inspire new research directions. With any self-similar group one can associate infinitely many finite graphs: the Schreier graphs of the action. The combinatorial properties of these graphs and their classification is not complete. One goal of this project consists in improving this partial classification. Moreover, Schreier graphs provide a new setting, in which physical and combinatorial models like the Ising model and the dimer model can be studied. Following strong indications coming from percolation theory, we inspect the algebraic properties of the groups by studying the physical properties of the model. Cellular automata (CA), classically studied on Euclidean grids, find in Schreier graphs a new setting in which can be defined. We would like to describe the property of CA in this new setting, find and extend results, already known in the context of Cayley graphs. Our last aim is to extend the lattices gas model to the Schreier graphs of self-similar groups. Achieving these targets would build a new bridge between computer science and group/graph theory.