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Topics in discrete mathematics

2019/2020
Programme:
Doctoral Programme Mathematics and Physics
Orientation:
Mathematics
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
ECTS:
6
Language:
slovenian, english
Lecturer (contact person):
Hours per week – 1. or 2. semester:
Lectures
2
Seminar
0
Tutorial
0
Lab
0
Content (Syllabus outline)

The content consists of a selection of standard topics in graduate-level discrete mathematics. Possible topics are, for example: symmetries of graphs, discrete geometry, algebraic and topological graph theory, configurations, extremal graphs, applications of graph theory in sciences and technology, symmetric functions, combinatorial species, advances enumeration methods. The choice depends on students' research interests.

Readings

Richard P. Stanley: Enumerative Combinatorics, Vol. 1 and 2, Cambridge University Press, New York-Cambridge, 1999, 2011.
R. Diestel: Graph Theory, 3rd ed., Springer, 2005.
J. A. Bondy, U. S. R. Murty: Graph Theory with Applications, 2nd ed., Springer, 2008.
I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford, University Press, 1995

Objectives and competences

The main goal of the course is to provide students with some important topics in discrete mathematics.

Intended learning outcomes

Knowledge and comprehension of presented concepts.
Ability to use acquired knowledge and skills.

Learning and teaching methods

Lectures, consultations, problem sessions

Assessment

Written exam (homeworks), oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Sandi Klavžar:
DORBEC, Paul, KLAVŽAR, Sandi. Generalized power domination: propagation radius and Sierpiński graphs. Acta applicandae mathematicae, ISSN 0167-8019, 2014, vol. 134, iss. 1, str. 75-86. [COBISS-SI-ID 17148249]
KLAVŽAR, Sandi. Structure of Fibonacci cubes: a survey. Journal of combinatorial optimization, ISSN 1382-6905, 2013, vol. 25, iss. 4, str. 505-522. [COBISS-SI-ID 16603737]
HAMMACK, Richard H., IMRICH, Wilfried, KLAVŽAR, Sandi. Handbook of product graphs, (Discrete mathematics and its applications). Boca Raton London New York: CRC Press, cop. 2011. XVIII, 518 str., ilustr. ISBN 978-1-4398-1304-1. [COBISS-SI-ID 15916121]
Matjaž Konvalinka:
KONVALINKA, Matjaž, PAK, Igor. Triangulations of Cayley and Tutte polytopes. Advances in mathematics, ISSN 0001-8708, 2013, vol. 245, str. 1-33. [COBISS-SI-ID 16706905]
KONVALINKA, Matjaž. Skew quantum Murnaghan-Nakayama rule. Journal of algebraic combinatorics, ISSN 0925-9899, 2012, vol. 35, no. 4, str. 519-545. [COBISS-SI-ID 16250713]
KONVALINKA, Matjaž. Divisibility of generalized Catalan numbers. Journal of combinatorial theory. Series A, ISSN 0097-3165, 2007, vol. 114, iss. 6, str. 1089-1100. [COBISS-SI-ID 14354265]
Marko Petkovšek:
ABRAMOV, Sergei A., BARKATOU, Moulay A., VAN HOEIJ, Mark, PETKOVŠEK, Marko. Subanalytic solutions of linear difference equations and multidimensional hypergeometric sequences. Journal of symbolic computation, ISSN 0747-7171, 2011, vol. 46, iss. 11, str. 1205-1228. [COBISS-SI-ID 16083033]
BRESSLER, Andrew, GREENWOOD, Torin, PEMANTLE, Robin, PETKOVŠEK, Marko. Quantum random walk on the integer lattice: examples and phenomena. V: AMS Special Sessions on Algorithmic Probability and Combinatorics, October 5-6, 2007, DePaul University, Chicago (Illinois), October 4-5, 2008, University of British Columbia, Vancouver (BC, Canada). LLADSER, Manuel (ur.), et al. Algorithmic probability and combinatorics : AMS special sessions on algorithmic probability and combinatorics, October 5-6, 2007, DePaul University, Chicago, Illinois, October 4-5, 2008, University of British Columbia, Vancouver, BC, Canada, (Contemporary mathematics, ISSN 0271-4132, 520). Providence: American Mathematical Society, cop. 2010, str. 41-60. [COBISS-SI-ID 15813977]
KLAVŽAR, Sandi, MOLLARD, Michel, PETKOVŠEK, Marko. The degree sequence of Fibonacci and Lucas cubes. Discrete Mathematics, ISSN 0012-365X. [Print ed.], 2011, vol. 311, iss. 14, str. 1310-1322. [COBISS-SI-ID 15884121]
Riste Škrekovski:
KNOR, Martin, LUŽAR, Borut, ŠKREKOVSKI, Riste. Sandwiching the (generalized) Randić index. Discrete applied mathematics, ISSN 0166-218X. [Print ed.], 2015, vol. 181, str. 160-166. [COBISS-SI-ID 2048330515]
KAISER, Tomáš, STEHLÍK, Matěj, ŠKREKOVSKI, Riste. Replication in critical graphs and the persistence of monomial ideals. Journal of combinatorial theory. Series A, ISSN 0097-3165, 2014, vol. 123, iss. 1, str. 239-251. [COBISS-SI-ID 16920665]
VUKAŠINOVIĆ, Vida, GREGOR, Petr, ŠKREKOVSKI, Riste. On the mutually independent Hamiltonian cycles in faulty hypercubes. Information sciences, ISSN 0020-0255. [Print ed.], jul. 2013, vol. 236, str. 224-235. [COBISS-SI-ID 26622247]