Generating, analysing and cataloging symmetric graph

The topic of the proposed project lies at the intersection of two mathematical disciplines: algebra and discrete mathematics, or more precisely, group theory and graph theory. It revolves around the concept of symmetry. It is spiced with computational flavours and intersects with trends in both data stewardship (such as the FAIR principles for scientific data) and mathematical knowledge management. Symmetry is a concept which plays a significant role in many areas of human activity. In mathematics, the need to understand symmetry gave birth to modern group theory, which may be seen as an attempt to understand the concept of symmetry in its purely abstract form. Groups are often studied in terms of their representations as symmetry groups of fixed mathematical objects, such as graphs. The study of groups acting on graphs have resulted in many deep theories and ground breaking results throughout mathematics (think of the Basse-Serre theory, the Classification of Finite Simple Groups, the theory Hurewitz groups and Rieman surfaces, the results of Gromov on locally compact topological groups etc). In all these endeavours, a crucial role play graphs of high level of symmetry. In discrete mathematics, enumeration of objects is a very important task. The lack of practical means of constructing lists of all small objects of a given class indicates the lack of our understanding of the theory. Attempts of constructing catagoues of graphs with high level of symmetry started in early 1930s, when Foster started collecting examples of arc-transitive graphs of valence 3. His work, now known as Foster's census, has been a valuable source of information for graph and group theorists for many decades. Several legendary mathematicians have been involved in constructions of catalogues of graphs of specific symmetry types, such as William Tutte, Harold Coxeter, John Conway etc. Every advancement of our theoretical understanding of symmetric graphs, together with ever increasing power of computers, spurs a new cycle of constructions of such catalogues. In return, new catalogues of highly symmetrical graphs spur further theoretical development. It is thus not surprising that catalogues of graphs are invaluable and one of the most cited resources in the area. This project represents one such cycle of understanding symmetry and its incarnation within the theory of graphs. On one hand, we aim to construct new catalogues of highly symmetrical graphs, significantly superseding existing and generating completely new ones. To this end, new tools and approaches will need to be invented. We thus strongly believe that pursuing the main goal of the project will motivate new research directions in certain areas of combinatorics and group theory, and thus increase general understanding of graphs with prescribed types of symmetry. We plan to use the obtained catalogue to search for patterns, testing existing and posing new conjectures, thus opening avenues for new relevant and meaningful theoretical research. The main objective of the proposed project can therefore be understood both as the final goal as well as the motivation and guideline for a more general research of symmetry properties of graphs, and symmetry in general. We shall be particularly mindful about the way the obtained datasets are stored and published. In particular, we plan to develop a web-based service for listing, browsing, storing, importing and exporting datasets of graphs, following the FAIR guiding principles for scientific data management (see, Scientific Data, 3.1, 2016).