DescriptionGraphs appear in many different contexts both pure and applied. Natural objects associated to a graph are the so-called graph Laplacians. Their spectra encode varies combinatorial properties of the underlying graphs. For example, the Colin de Verdiere spectral graph invariant can be used to characterise global graph properties like planarity. One motivation to study this invariant was to find a proof of the famous Four Colour Theorem. Another thriving area of practical importance is to find explicit geometric realisations of graphs in Euclidean space and to split the graph vertices into different clusters with very few edges between them. These geometric realisations are obtained via the eigenvectors of the graph Laplacians and are called ''spectral representations''. The quality of the associated spectral clustering can be theoretically justified by relations between eigenvalue gaps and Cheeger isoperimetric constants. In this project, our first aim is to become acquainted with the general area of graphs and their Laplacians. Then each student will specialise into a different topic and read and study original research articles.PrerequisitesThere is no explicit prerequisite, but it is useful if the participants took the course Analysis III in 2013/14.ResourcesSome recommendable references in this topic are
email: N Peyerimhoff |