## Division of Discrete Mathematics

**Chair:**

Assoc. Prof. Vedrana Mikulić Crnković

**Members:**

- Prof. Dean Crnković
- Prof. Sanja Rukavina
- Assoc. Prof. Vedrana Mikulić Crnković
- Assist. Prof. Andrea Švob
- Assist. Prof. Doris Dumičić Danilović
- Assist. Prof. Marija Maksimović
- Dr. Sara Ban
- Dr. Daniel Hawtin
- Dr. Nina Mostarac
- Dr. Marina Šimac
- Ana Grbac
- Matteo Mravić
- Ivona Novak
- Tin Zrinski
- Matea Zubović

**Science**

Division of Discrete Mathematics was formed in early 2011. The division consists of 15 members, six of which are holding tenure track positions, four senior teaching assistants and five teaching assistants.

Scientific work of the members of the Division is in the area of combinatorics and discrete mathematics, especially design theory, graph theory and coding theory.

Active research projects:

- Combinatorial Objects and Codes (principal investigator: Dean Crnković) financed by the Croatian Science Foundation
- Codes, groups and combinatorial structures (principal investigator: Dean Crnković)
- Designs, graphs and linear codes (principal investigator: Sanja Rukavina)
- Permutation groups as designs, geometric designs and finite geometries(principal investigator: Vedrana Mikulić Crnković)
- Designs, codes, graphs and cryptography - interdisciplinary approach to discrete structure analysis (principal investigator: Sanja Rukavina), bilateral project Načrti, kode, grafi in kriptografija interdisciplinarni pristop v analizi določenih diskretnih struktur (Univerza na Primorskem, Inštitut Andrej Marušič - principal investigator Enes Pasalic).

- Codes and Related Combinatorial Structures (principal investigator: Dean Crnković), project financed by the Croatian Science Foundation (2014-2018)
- Block designs, strongly regular graphs and related combinatorial structures (principal investigator: Dean Crnković), support for research at the University of Rijeka (2013-2018)
- Permutation groups as designs, geometric designs, codes and finite geometries (principal investigator: Vedrana Mikulić Crnković), initial support for research at the University of Rijeka for young researchers (2017)
- Block designs, strongly regular graphs and related combinatorial structures (principal investigator: Dean Crnković), financed by Ministry of science, education and sports of the Republic of Croatia

In 2010, members of the Division participated in the organization of NATO Advanced Study Institute on Information Security and Related Combinatorics summer school (main sponsor: NATO; Organizing Committee: Dean Crnković; Willem Haemers, Tilburg University; Sanja Rukavina, Vladimir D. Tonchev, Michigan technological University).

Members of the Division are actively involved in the international scientific community, collaborating with mathematicians from different countries and participating at international scientific conferences. We are especially proud that the article *Self-dual codes from quotient matrices of symmetric divisible designs with the dual property* (D. Crnković, N. Mostarac, S. Rukavina) is in the list of articles Editors' Choice from Discrete Mathematics.

Selected research papers:

- D. Crnković, R. Egan, A. Švob, Constructing self-orthogonal and Hermitian self-orthogonal codes via weighing matrices and orbit matrices, Finite Fields Appl. 55 (2019), 64-77.
- M. De Boeck, M. Rodgers, L. Storme, A. Švob, Cameron-Liebler sets of generators in finite classical polar spaces, J. Combin. Theory Ser. A 167 (2019), 340-388.
- R. Egan, P. Ó Catháin, Morphisms of Butson classes, Linear Algebra Appl. 577 (2019), 78-93. R. Egan, Phased unitary Golay pairs, Butson Hadamard matrices and a conjecture of Ito's, Des. Codes Cryptogr. 87 (2019), 67-74.
- D. Crnković, D. Dumičić Danilović, S. Rukavina, M. Šimac, On some new Steiner 2-designs S(2,5,45), Util. Math. 111 (2019), 281-308.
- D. Crnković, N. Mostarac, Self-dual codes from orbit matrices and quotient matrices of combinatorial designs, Discrete Math. 341 (2018), 3331–3343.
- D. Crnković, S. Rukavina, A. Švob, New strongly regular graphs from orthogonal groups O+(6,2) and O-(6,2), Discrete Math. 341 (2018), 2723-2728.
- D. Crnković, V. Mikulić Crnković, B. G. Rodrigues, On self-orthogonal designs and codes related to Held's simple group, Adv. Math. Commun. 12 (2018), 607-628.
- D. Crnković, R. Egan, A. Švob, Orbit matrices of Hadamard matrices and related codes, Discrete Math. 341 (2018), 1199-1209.
- D. Crnković, V. Mikulić Crnković, A. Švob, New 3-designs and 2-designs having U(3,3) as an automorphism group, Discrete Math. 340 (2017), 2507-2515.
- D. Crnković, B. G. Rodrigues, S. Rukavina, V. D. Tonchev, Quasi-symmetric 2-(64,24,46) designs derived from AG(3,4), Discrete Math. 340 (2017), 2472-2478.
- D. Crnković, M. Maksimović, B. G. Rodrigues, S. Rukavina, Self-orthogonal codes from the strongly regular graphs on up to 40 vertices, Adv. Math. Commun. 10 (2016), 555-582.
- R. Egan, D. L. Flannery, Automorphisms of generalized Sylvester matrices, Discrete Math. 340 (2017), 516-523.
- M. De Boeck, L. Storme, A. Švob, The Cameron-Liebler problem for sets, Discrete Math. 339 (2016), 470-474.
- D. Crnković, S. Rukavina, Self-dual codes from extended orbit matrices of symmetric designs, Des. Codes Cryptogr. 79 (2016), 113-120.
- D. Crnković, N. Mostarac, S. Rukavina, Self-dual codes from quotient matrices of symmetric divisible designs with the dual property, Discrete Math. 339 (2016), 409-414.

Members of the Division participate in the implementation of the Croatian Doctoral Program in Mathematics (joint program of J.J. Strossmayer University of Osijek, University of Rijeka, University of Split, University of Zagreb) organizing Finite Mathematics Seminar. As a part of the Seminar, members of the Division present their own research work as well as other relevant articles in the area of research.

We point out that Andrea Švob has won the University of Rijeka Foundation Award for year 2015 in the category young scientists - natural and technical sciences and that Vedrana Mikulić Crnković whas awarded the same for the academic year 2008/2009.

**Teaching**

Since the academic year 2011/2012 the Department has been organizing and running graduate studies Discrete Mathematics and Its Applications. Members of the Division have designed and run courses in the area of discrete and combinatorial mathematics in these studies and work together on designing and running each course for the purpose of improving the quality of the studies and greater correlation between the courses. The Division is the executor of the project within which e-materials have been made for the course Introduction to Design Theory which has been financed by University of Rijeka (through the Tender for financing the development of e-courses in the academic year 2011/2012).

In 2012, the Laboratory for Discrete Mathematics was formed and equipped (with GAP and Magma software) and has since been used for research work by the members of the Division and for teaching purposes of the graduate studies Discrete Mathematics and Its Applications.

We point out that the members of the Division are course executors of two courses that have been awarded for special contribution for development of e-learning at the University of Rijeka: course ICT in Mathematics (academic year 2009/2010) and course Geometry (academic year 2010/2011). Furthermore, four members of the Division have won the University of Rijeka Teaching Excellence Award (Ivona Novak and Andrea Švob in academic year 2015/2016 and Sanja Rukavina and Marina Šimac in academic year 2017/2018).

Members of the Division are authors and executors of multiple popular lectures and workshops which are designed for the elementary and high school audiences for the purpose of popularization of mathematics and mathematical studies. They also actively participate in the work of associations sucha as: Mathematicians and Physicists Society, Alumni Club MatRI, Zlatni rez, Universitas,...

**Professional activities**

In addition to science, members of the Division are actively engaged in professional work.

Selected professional activities:

- Members of the Division participate as lecturers in the activities of the County Expert Councils and as invited lecturers at meetings of mathematics teachers.
- S. Rukavina was a member of the Organizing Committee of the Festival of Science from 2002 to 2012 and V. Mikulić Crnković has been a member since 2013.
- S. Rukavina, M. Žuvić-Butorac, J. Ledić, B. Milotić, R. Jurdana-Šepić, Developing positive attitude towards science and mathematics through motivational classroom experiences, Science Education International, Vol.23, No.1 (2012) 6-19
- Mathematics and ICT, Varaždin, 19-20 September 2012, presentation: Examples of the use of LMS in mathematics teaching at the Department of Mathematics, University of Rijeka (authors: V. Mikulić Crnković and S. Rukavina)
- D. Crnković, Some applications of discrete mathematics (public lecture at the Festival of Science 2011)
- V. Mikulić, Golden ratio (public lecture at the Festival of Science 2009)
- S. Rukavina, R.Jurdana-Šepić, Changes in the Croatian educational system - the first Steps, 2008, International Journal of Research in Education, Vol. 1, No. 1, 2009, 1-12.
- S. Rukavina was the leader of the project Development of natural science and mathematics literacy through active learning (executor: Association Zlatni rez, partner: Department of Education, City of Rijeka) financed by the Ministry of Science, Education and Sports from 2007 to 2010. A. Grbac, D. Dumičić Danilović, V. Mikulić Crnković and A. Švob have also participated in this project as authors and workshop leaders.
- S. Rukavina is the author of je autor of several radio shows "Baltazar“ with topics in mathematics.

Members of the Division include students in professional work:

- K. Jurjak, A. Švob, Graphs in LaTeX, Mathematics and school (2018), 93
- K. Jurjak, A. Švob, Program LaTeX - Writing mathematical text, Part 2, Mathematics and school (2017), 92
- K. Jurjak, A. Švob, Program LaTeX - Writing mathematical text, Part 1, Mathematics and school (2017), 88
- Tihana Sabo, Sanja Rukavina, Origami and second-order curves, Mathematics and school, 86 (2017)
- David Bojanić, Andrea Švob, Two classical geometrical problems, Matka 25, (2016./2017.) br. 99
- Mia Černivec, Sanja Rukavina, Workshop "Combinatorial games", Mathematics and school, 61 (2014)
- Sara Ban, Sanja Rukavina, Latin Squares, Matematičko-fizički list, 4, 2011/2012.
- Vedrana Mikulić Crnković, Ivona Novak, On the models of the hyperbolic plane, Croatian mathematical electronic journal math.e 22 (2012)
- Sanja Rukavina, Marija Sekulić, Andrea Švab, Sanja Vranić, Workshop "Help Joseph Kruskal!", Mathematics and school, 45 (2008)
- Milan Bakić, Dina Kovačević, Sanja Rukavina, Why is there a shortage of mathematics teachers? High school students' attitudes about the profession "teacher of mathematics", Third international scientific colloquium Mathematics and child, Osijek, monograph (2011)