Advances in Spectral Graph Theory and Its Applications

Dear Colleagues,

Graph data structures generalize the regularity of time series and image data structures and provide opportunities for the study of propagations and dynamic phenomena on regular and irregular networks and hypergraphs with higher dimensional topological building blocks. The general idea of topological data analysis is effectively tied to the main concepts of spectral graph theory, at least if one considers utilizing the spectra of the graph operators associated with the topological structure of the data. Today, beyond the classical definition of the field of spectral graph theory by pioneers such as Prof. Fan Chung, the interdisciplinary domain of spectral graph theory is explicitly or implicitly present in various areas, and it is especially of note in hybrid applications with artificial intelligence. The field has already expanded beyond the properties of simple graphs into the study of hypergraphs (meshes) in geometry processing and shape learning. With the advents of graph neural networks and geometric deep learning, the continued development of the field of spectral graph theory is more relevant than ever. There are at least five areas of study that will hopefully intertwine in this Special Issue, namely the concept of signal processing on graphs and hypergraphs with differential and integral operators, the study of Markov chains and Markov decision processes through their spectra, spectral geometry processing, shape learning and dimensionality reduction in machine learning and deep learning, and the physical study of vibrations in acoustics, structural engineering, and dynamical systems through the spectra of the vibrating material manifolds.

It is hard to find a part of mathematics with such great connective potential for a wide array of applications in science and engineering comparable to that of spectral graph theory. The algebraic language of the field makes it particularly suitable for modern computational applications and the ultimate vectorization of algorithms for GPU computing and AI applications. Nevertheless, these rapid expansions lead to the different frontiers not being necessarily synchronized with the state of the art in the mathematical field of spectral graph theory; furthermore, the connections between these diverse applications through the core of the theory, i.e., the discretization of multi-variate differential and integral operators on graphs, are not widely discussed or acknowledged.

The purpose of this Special Issue is to contribute to the increasing cohesion in this umbrella field by providing a collection of articles that ideally relate the theory and practice of two or more of the multiple application areas of spectral graph theory, namely the study of Markov chains and Markov decision processes, shape learning, geometric deep learning, dimensionality reduction and manifold learning, graph signal processing, topological data analysis, spectral methods [for solving PDEs], as well as physics and engineering applications related to the study of vibrations of manifolds (such as acoustics and structural engineering). 

Dr. Pirouz Nourian
Guest Editor

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