Research group for combinatorial and discrete mathematics

• Tanja Vučičić, retired

Research areas:

• Discrete mathematics and Combinatorics
• Design theory

Research description:

Our research is focused on various kinds of combinatorial structures such as t-designs, symmetric designs, difference sets, partial difference sets, strongly regular graphs and flag-transitive incidence structures. It includes investigating their properties and links, their construction and classification.

The way we approach the classification problem is through imposing different conditions on the action of finite automorphism groups of the structures under observation. One such condition is transitivity of the group action, in particular: point-transitivity, block-transitivity, flag-transitivity or multiple transitivity. A transitive permutation group belongs to either class: primitive or imprimitive. We consider the action of groups from both classes. In addition to theoretical aspect, for computing with permutation groups we use systems such as GAP and MAGMA.

We have developed new construction method for symmetric designs, difference sets and flag-transitive incidence structures.

Contacts with academic and other institutions:

– Department of Mathematics, University of Rijeka
– PMF – Department of Mathematics, University of Zagreb
– Faculty of Electrical Engineering and Computing, University of Zagreb

Description of the current cooperation:

Members of this research group and a number of researchers in the same area from the listed institutions have organized mutual visits and given many talks, both as guests and as hosts, under the framework of their related seminars. They have collaborated in different research and in proposing scientific projects in the area of combinatorial and discrete mathematics.

Plans for future research:

The research will include developing the theory of flag-transitive designs, interesting from the geometrical point of view for admitting a comparatively large automorphism group. The emphasis will be on imprimitive automorphism group action. An improvement of algorithms for the design construction is expected. The intention is to show the application of the obtained algorithms in the construction procedure of concrete and specific designs. Symmetric flag-transitive designs are of special interest.

We will approach several aimed classifications of the flag-transitive designs (that we have in view on the trace of some open questions) through flag-transitive incidence structures and edge-transitive bipartite graphs. Moreover, in some problems edge-transitive graphs are expected to appear.

List of selected research papers:

1. Braić, A. Golemac, J. Mandić and T. Vučičić, Primitive Symmetric Designs with Prime Power Number of Points, Journal of Combinatorial Designs, 18, (2010), 141-154.
2. Braić, A. Golemac, J. Mandić and T. Vučičić, Graphs and Symmetric designs corresponding to difference sets in groups of order 96, Glasnik matematički, Vol. 45 (65) (2010), 1-14.
3. Braić: Primitive Symmetric Designs with at most 255 Points, Glasnik matematički Vol. 45 (65), No. 2 (2010), 291-305.
4. Braić, A. Golemac, J. Mandić and T. Vučičić, Primitive Symmetric Designs with up to 2500 Points, Journal of Combinatorial Designs, 19, (2011), 463-474.
5. Mandić, M. O. Pavčević and K. Tabak, On difference sets in high exponential 2-groups, Journal of Algebraic Combinatorics 38 (2013), 785-795.
6. Braić, J. Mandić and T. Vučičić, Primitive Block Designs with Automorphism Group PSL(2,q), Glasnik matematički, Vol. 50, No.1 (2015), 1-15.
7. Mandić and T. Vučičić, On the existence of Hadamard difference sets in groups of order 400, Advances in Mathematics of Communications, Vol. 10, No. 3 (2016), 547-554.
8. Braić, J. Mandić and T. Vučičić, Flag-transitive block designs with automorphism group S-n wr S-2, Discrete mathematics, 341 (2018), 8; 2220-2230.
9. Vučičić, Hadamard difference sets and related combinatorial objects in groups of order 144, Rad Hrvatske akademije znanosti i umjetnosti, Razred za matematičke, fizičke i kemijske znanosti. Matematičke znanosti, 23 (2019), 13-29.