UP FAMNIT successful with three new research projects

Title: Automorphisms and Isomorphisms of Finite Graphs
Principal investigator: Prof. István Kovács, PhD

Combinatorial structures with a high level of symmetry are often explored in applications to both natural and social sciences. In some of these applications there is a demand to compare two objects effectively. A mathematical model capturing this situation is a finite graph with non-trivial symmetries, and the underlying mathematical discipline is algebraic graph theory (AGT).

The symmetry of a graph is measured by its automorphism group and the alikeness of objects is expressed by the concept of a graph isomorphism.

In the core of ATIFG lies the Cayley isomorphism problem, a well-known and long-standing open problem in AGT. In the proposal we outline several (sub)-problems, which will serve as mile-stones in the final solution.

Title: Tools for innovative drug design
Principal investigator: Prof. Dušanka Janežič, PhD

This project is at the forefront of current research trends in the field of molecular modeling. We will develop new molecular modeling methods based on chemical graph theory, a branch of mathematical chemistry dealing with discrete structures in chemistry.

A special focus will be on the development of new algorithms for the prediction of protein binding sites (ProBiS) and new web tools for modeling of pharmaceutically interesting molecules - ProBiS Tools (algorithm, database, web server).

The ProBiS Tools will be the first to allow the identification of interactions between protein structures, the prediction of ligand selectivity and binding, and the monitoring of the effects of conserved waters and sequence variants on ligand binding, to surpass human involvement in drug design.

We aim to develop a free web-based protein interaction atlas to predict genetic variations involved in drug interactions and disease development - ProBiS-ATLAS of all available protein binding sites in the PDB with mapped ligands and sequence variants. The atlas will also include information on the effects of sequence variants on binding for each specific ligand in the PDB. This newly developed approach will be particularly useful in the context of precision medicine.

Our ProBiS Tools will make it possible to combine several otherwise unrelated fields of research, i.e. graph theoretical approaches, genome sequence studies, protein structures and molecular dynamics simulation.
Despite the great potential for the concrete application of our results in specific technological and industrial sectors, the main focus of our research is still on the development of general, new mathematical methods and algorithms in the field of molecular modeling, and thus represents a contribution to the overall scientific knowledge.

The project involves a number of researchers with excellent publishing records, currently working in Slovenia guaranteeing research at the highest possible level.

Title: Graph Theory in Preserver Problems
Principal investigator: Prof. Bojan Kuzma, PhD

The proposed project is roughly divided into 4 parts and belongs to the theory of graphs induced by relations and general preservers on:

• normed spaces,
• Hilbert C*-modules,
• matrix algebras,
• sets of (possibly unbounded) operators, and
• manifolds related to special relativity.

In particular we aim to:

1. Study the properties of graph induced by Birkhoff-James and Roberts orthogonality and also induced by strong Birkhoff-James orthogonality and study their (linear) preservers.
2. Classify general bijective bi-preservers of left-star partial order on B(H).
3. Classify additive maps which preserve the spectrum of unbounded linear operators on a complex Banach space.
4. Classify maps on the de Sitter space, which maps light-like geodesics into light-like geodesics. Injectivity of the maps will be assumed only on each light-like geodesic separately.