Izredno veliki grafi in omrežja (Slovene)

Our main concern will be the study of very large graphs, the main theme being the relationship between local and global properties. There is need for basic theory, although the case of dense graphs (Lovasz, Szegedy, Borgs, Chayes, Sos, Vesztergombi) and, at the other extreme, the case of planar graphs (Schramm, Angel, LeGall, Marckert in Mokkadem) are well understood. Our goal is to discover and use connections between analytic, algebraic and topological properties of large graphs. The main themes are convergence and limits of graphs embedded in surfaces, graphs in other minor-closed families, application of nonstandard analysis (Gromov-Hausdorff metric) for comparison of large objects, theory of probabilistic algorithms based on local observations, study of products and bundles and development of fundamental algorithms for large graphs.

The project belongs to basic research from the area of mathematics. Problems that we work on are internationally important, which can in particular be justified with our bibliography from the last period as well as with the (citation) impact of our results. The problems are central in the area of modern graph theory. Our newly obtained results have been published in established international journals and are presented at international scientific conferences. For the next period we expect that we will be invited to deliver several invited plenary talks which will further emphasize the visibility of our research achievements. In this way we will further increase the role of the Slovenian graph theory school.

The results obtained are mostly theoretical. Nevertheless, some of our results have a potential for applications which is in particular the case with our research in algorithmic and optimization aspects of graph theory. The area of "big data" and hence also our main theme of large graphs is one of the priorities listed in the research policy documents worldwide.