Laplacian Operators for Graphs

Discrete Laplacian operators appear in the mathematical description of the majority of dynamical processes occurring on these systems. I am interested in the definition, mathematical analysis and application of new Laplacian operators for graphs/networks and their potential extension to hypergraphs, multiplexes, simplicial complexes and metaplexes. I have made a generalization of the discrete Laplacian operator to account for long-range hops in graphs/networks. These d-path Laplacian operators and their transformation using Mellin transform have proved to describe qualitatively new physical processes such as superdiffusion. I have also interest in the development and application of hubs-repelling and hubs-attracting Laplacian operators to describe processes in which the most connected nodes of the graphs/networks either repel or attract the diffusive particle. This research is financially supported by a Grant from Ministerio de Ciencia, Innovacion y Universidades of Spain. Another area of research in this field is the study of fractional differential equations for graphs/networks to describe either sub- or superdiffusive processes as well as the combination of discrete and continuous operators of this kind in metaplexes.