GRAPH reconstruction, COspectrality and SYnchronisation through the lens of number theory, geometry and algorithms

Graph theory is a field with many beautiful and powerful connections to other areas of mathematics and computer science. In this project, the experienced researcher (ER), an expert in graph theory, will further explore these connections together with the experts in number theory, manifold and parameterised complexity from the host. Moreover, solving the chosen problems will improve our understanding of several famous open problems.- Graph reconstruction (WP1): does local information of the graph (e.g. subgraph counts) determine graph invariants (e.g. the number of spanning trees)? The ER will continue her excellent track record in this area with the help of the number theoretical expertise of the host. - Graph isomorphism (WP2): study small common covers of pairs of graphs, building up to the computation of a set of matrices whose spectra determine whether the graphs are isomorphic. This package carries concepts and techniques within the expertise of the supervisor to graph theory, opening up a new avenue of research.- Černý's conjecture (WP3): probabilistic and parameterised complexity aspects are studied of the question `How long is the shortest reset word in a given finite state automaton?'. This combines the expertise of the ER in extremal and probabilistic combinatorics with the expertise in parameterised complexity of the host.