Spectrum of generalized Petersen Graphs

The Australasian Journal of Combinatorics 49 (2011), 39-45In this paper, we completely describe the spectrum of the generalized Petersen graph P(n/k), thus adding to the classes of graphs whose spectrum in completely known

On the Falk invariant of hyperplane arrangements attached to gain graphs

The fundamental group of the complement of a hyperplane arrangement in a complex vector space is an important topological invariant. The third rank of successive quotients in the lower central series of the fundamental group was called Falk invariant of the arrangement since Falk gave the first formula and asked to give a combinatorial interpretation. In this article, we give a combinatorial formula for the Falk invariant of hyperplane arrangements attached to certain gain graphs.Comment: To appear in the Australasian Journal of Combinatorics. arXiv admin note: text overlap with arXiv:1703.0940

Young classes of permutations

We characterise those classes of permutations having the property that for every tableau shape either every permutation of that shape or no permutation of that shape belongs to the class. The characterisation is in terms of the dominance order for partitions (and their conjugates) and shows that for any such class there is a constant k such that no permutation in the class can contain both an increasing and a decreasing sequence of length k.Comment: 11 pages, this is the final version as accepted by the Australasian Journal of Combinatorics. Some more minor typos have been correcte

Counting the spanning trees of the 3-cube using edge slides

We give a direct combinatorial proof of the known fact that the 3-cube has 384 spanning trees, using an "edge slide" operation on spanning trees. This gives an answer in the case n=3 to a question implicitly raised by Stanley. Our argument also gives a bijective proof of the n=3 case of a weighted count of the spanning trees of the n-cube due to Martin and Reiner.Comment: 17 pages, 9 figures. v2: Final version as published in the Australasian Journal of Combinatorics. Section 5 shortened and restructured; references added; one figure added; some typos corrected; additional minor changes in response to the referees' comment

Poolrühmade Cayley graafid

Bakalaureusetöös vaadeldakse poolrühmade Cayley graafe. Peamine eesmärk on kirjeldada tingimusi, mille korral poolrühma Cayley graaf on mittesuunatud. Põhitulemustes vaadeldakse neid nõudeid eraldi perioodiliste ja lõplike poolrühmade korral. Teema ilmestamiseks on lisatud terve rida näiteid, sealhulgas ka mitteperioodiliste poolrühmade kohta, ning esitatakse võrdluseks suunatud Cayley graafe. Töö baseerub A.V. Kelarevi teadusartiklil On undirected Cayley graphs (Australasian Journal of Combinatorics 25, 2002, 73–78)

Perfect countably infinite Steiner triple systems

We use a free construction to prove the existence of perfect Steiner triple systems on a countably infinite point set. We use a specific countably infinite family of partial Steiner triple systems to start the construction, thus yielding 2ℵ0 non-isomorphic perfect systems