Projects / Programmes
Bipartite distance-regular graphs
| Code |
Science |
Field |
Subfield |
| 1.01.05 |
Natural sciences and mathematics |
Mathematics |
Graph theory |
| Code |
Science |
Field |
| P001 |
Natural sciences and mathematics |
Mathematics |
| Code |
Science |
Field |
| 1.01 |
Natural Sciences |
Mathematics |
Distance-regular graphs; Q-polynomial property; Terwilliger algebra; uniform partialy ordered sets; group actions on graphs
Researchers (8)
Organisations (2)
Abstract
Our research concerns a combinatorial object known as a graph. A graph is a finite set of vertices, together with a set of undirected edges, each of which connects a pair of distinct vertices. We say that vertices x, y are adjacent whenever x,y are connected by an edge. The concept of a graph is useful because mathematical as well as intuitive notions can be formulated in terms of adjacency.
Our research concerns a type of graph said to be distance-regular. A connected graph is distance-regular, if the cardinality of the intersection of two spheres depends only on the radii and the distance between the centres of these two spheres. The 1-skeletons of the five platonic solids provide examples of distance-regular graphs. The theory of distance-regular graphs is connected to some other areas of mathematics, such as coding theory, representation theory, and the theory of orthogonal polynomials.
The main goal of our research is to understand and classify bipartite Q-polynomial distance-regular graphs. To reach this goal we will study the so-called Terwilliger algebra T of G. In many instances algebra T is studied via its irreducible mudules. In the study of irreducible T-modulesles, the so-called endpoint of an irreducible T-module plays important role. Morover, among irreducible T-modules, thin T-modules are of particular interest for us. In the course of the project we will try to solve the following problem:
Classify those bipartite distance-regular graphs G for which up to isomorphism there exist at most two irreducible T-modules with endpoint 2, and they are both thin.
We strongly suspect that such graphs are closely related to the bipartite Q-polynomial distance-regular graph. Therefore, the solution of the above problem would help us to better understand bipartite Q-polynomial distance-regular graph. We will also try to solve a series of related problems.
Part of our research will also be the investigation of group action on distance-regular graphs. In particular, we will try to classify Cayley distance-regular graphs (for a various classes of groups) and bi- and tri-Cayley distance-regular graphs (again for a various classes of groups).
Significance for science
Apart from Art, Mathematics is the only universal language of human communication present in all civilizations. Abstract mathematical theories are used in Natural Sciences, Engineering, Computer Science and also in Social, Economic and Biomedical Sciences. It has an essential role in many important areas of research, such as Safe Communications, Data Protection, or Decoding of Humane Genome, thus proving that its influence to the very foundations of modern society has reached previously unthinkable levels. The project was at the cutting edge of today's research in the theory of distance-regular graphs. Results of the project will be of great importance in the effort of researchers to understand (classify) Q-polynomial bipartite distance-regular graphs and Cayley distance-regular graphs, as well as generalizations of these.
Significance for the country
These stormy times of social changes call for an even tighter incorporation of Mathematics into scientific research and education thus enabling a faster technological development in Slovenia. Our group already has many experiences in this field. Project is important for Slovenia because of the following reasons. (1) The project group is included into a grant awarded by the European Commission under the Horizon 2020 Teaming grant instrument. The purpose of the funds is to increase innovation excellence in Europe in general, especially in member states underperforming in innovation. (2) Members of the project team are widely recognized within the researchers, mainly becouse of their excelent results and international conferences that they are organizing. This, of course, implies also better recognition of slovenian science in general. (3) The results of the project increase the research performance of University of Primorska in general. Because of its geographic position, University of Primorska is of great importance for Slovenia. Our existence as a fully developed nation in Europe depends as much on preserving our language and culture as it depends on having a highly educated population. It is thus necessary to be able to use different communication channels – and mathematics, as a universal language, is a key factor here.
Most important scientific results
Annual report
2011,
2012,
2013,
final report,
complete report on dLib.si
Most important socioeconomically and culturally relevant results
Annual report
2011,
2012,
2013,
final report,
complete report on dLib.si
