#### The Study of Complex Systems at Different Size Scales

Our research is directed to the understanding of the structure, dynamics and functioning of complex systems at different size-scales. Our philosophy is that it is more important to understand the organizational principles of such systems as a whole than understanding the role of their individual components. Then, we consider an integration of complex systems through the use of a common representation by means of graphs, which are currently known as complex networks.

#### Research Areas

#### Mathematics of Networks, Hyper-Networks and Multiplex Networks

Our interest is in developing new mathematical concepts that permit us to understand the organization and function of complex systems. We combine techniques and tools from Graph Theory, Linear Algebra and Statistical Mechanics to characterise the structural and dynamical properties of networks, hypernetworks and multiplexes. In particular, we are interested in the use of matrix functions, operator theory, geometric embedding, and algebraic topology, among others to characterise properties such as expansibility, topological and functional bottlenecks, organization of clusters, global communicability, “clumpiness”, returnability, embedding into Euclidean spaces, random rectangular graphs, etc. In addition, we study dynamical properties of networks, hypernetworks and multiplexes such as diffusion processes, synchronization and epidemic spreading and relate these dynamical properties to the structural characteristics of these networks via algebraic graph theory.

#### Applications of Networks to Real-World Problems

We are interested in the applications of network theory to solve problems in a variety of real-world scenarios. These applications currently include: biomolecular networks, such as protein-protein interaction and metabolic networks; infrastructural networks, such as urban street and airline transportation networks; ecological networks, such as food webs and landscape systems; fracture networks in rocks and the diffusion of oil and gas on them; social networks and the influence of direct and indirect peer pressure on the diffusion of information; networks that change in time, such as telecommunication networks; deployment of wireless sensor networks in geographical areas; etc.

#### Mathematical Chemistry

We are currently interested in the generation of algebraic invariants to represent molecular structures, the generalization of graph theoretic invariants, the study of spectral properties of molecular graphs and the elaboration of a theory combining the use of graph theory, quantum mechanics and statistical mechanics to understand the molecular structure. These methods include perturbative theory based on simple tight-binding Hamiltonian and the use of matrix functions of such Hamiltonians. In addition, we are also interested in the design of new molecular structural with unusual properties, mainly based on carbon-only structures.