2014 Phd Summer School in Discrete Mathematics and Symmetries of Graphs and Networks IV

Contribution List

7. 2-Arc-Transitive Metacyclic Covers of Complete Graphs

Dr Wenqin Xu (Capital Normal University)

Regular covers of complete graphs whose fibre-preserving automorphism groups act 2-arc-transitively are investigated. Such covers have been classified when the covering transformation groups $K$ are cyclic groups $\text{Undefined control sequence ZZ}$ for an integer $d\ge 2$, metacyclic abelian groups $\text{Undefined control sequence ZZ}$, or nonmetacyclic abelian groups $\text{Undefined control sequence ZZ}$ for a prime $p$ (see S.F. Du, D. Maru\v si\v c and A.O. Waller, On...

8. A classification of regular embeddings of graphs of order prime-cube

Prof. Shaofei Du (Capital Normal University)

SYGN Invited Talk

A 2-cell embedding of a graph into an orientable or nonorientable closed surface is called regular if its automorphism group acts regularly on its arcs and flags respectively. One of central problems in topological graph theory is to classify regular maps by given underlying graphs or automorphism groups. In this talk, we shall introduce a classification of the regular embeddings of...

1. Automorphisms of Cayley graphs that respect partitions

Prof. Joy Morris (University of Lethbridge)

Summerschool Lecture

A Cayley graph $\mathrm{\Gamma }=\mathrm{C}\mathrm{a}\mathrm{y}(G;S)$ on a group $G$ with connection set $S$, is a graph whose vertices are labelled with the elements of $G$, with vertices $g_1$ and $g_2$ adjacent if $g_1^{-1}{g}_2\in S$. We say that an automorphism $\alpha$ of $\mathrm{\Gamma }$ respects the partition $\mathcal{C}$ of the edge set of $\mathrm{\Gamma }$ if for every $C\in \mathcal{C}$, we have $\alpha (C)\in \mathcal{C}$. I will...

5. Billiard Arrays and finite-dimensional irreducible Uq(sl2)-modules

Prof. Paul Terwilliger (University of Wisconsin-Madison, USA)

SYGN Invited Talk

In this talk we will describe the notion of a Billiard Array. This is a triangular array of one-dimensional subspaces of a finite-dimensional vector space, subject to several conditions that specify which sums are direct. We use Billiard Arrays to characterize the finite-dimensional irreducible Uq(sl2)-modules, for q not a root of unity. The equitable presentation of Uq(sl2) comes up naturally...

3. Classification of reflexible edge-transitive embeddings of $K_{m,n}$ and corresponding groups

Prof. Young Soo Kwon (Yeungnam University)

SYGN Invited Talk

In this talk, we classify reflexible edge-transitive embeddings of complete bipartite graphs by partite set preserving automorphisms. As a by-product, we classify groups $\mathrm{\Gamma }$ such that {\rm (i)} $\mathrm{\Gamma }=XY$ for some cyclic groups $X=⟨x⟩$ and $Y=⟨y⟩$ with $X\cap Y=\{1_{\mathrm{\Gamma }}\}$ and {\rm (ii)} there exists an automorphism of $\mathrm{\Gamma }$ which sends $x$...

13. Codes constructed from orbit matrices of block designs

Dr Loredana Simčić (Faculty of Engineering, University of Rijeka)

In this talk we will describe under which conditions the orbit matrix of a block design under the action of automorphism group generates a self-orthogonal code. The obtained results are the generalization of the construction method of self-orthogonal codes from orbit matrices of block designs under the action of an automorphism group of prime order with no fixed points and blocks, given by...

23. Colorings of affine and projective spaces

Prof. György Kiss (Eötvös Loránd University, Hungary)

SYGN Invited Talk

There are sevaral variations of vertex and edge colorings of hypergraphs. A {\em vertex coloring\/} of a hypergraph $\mathcal{H}=(X,\mathcal{C})$ is a mapping $\varphi$ from $X$ to a set of colors $\{1,2,\dots ,k\}.$ A {\em strict\/ rainbow-free $k$-coloring\/} is a mapping $\varphi :X\to \{1,\dots ,k\}$ that uses each of the $k$ colors on at least one vertex such that each edge of $\mathcal{H}$ has at...

21. Compatible elements for a tridiagonal pair

Gabriel Pretel (University of Wisconsin-Madison, USA)

Roughly speaking, tridiagonal pairs of Krawtchouk type correspond to the finite-dimensional irreducible modules of a certain Lie algebra known as the Onsager algebra. E. Date and S. S. Roan showed that the Onsager algebra is embedded in another Lie algebra known as the sl2 loop algebra. They classified the finite-dimensional irreducible modules for the Onsager algebra, and in particular they...

20. Finite-dimensional irreducible modules for an even subalgebra of $U_q({\mathfrak{sl}}_2)$.

Alison Gordon Lynch (University of Wisconsin-Madison)

The quantum algebra $U_q({\mathfrak{sl}}_2)$ has connections to $Q$-polyonomial distance-regular graphs, tridiagonal pairs of linear transformations, the $q$-tetrahedron algebra, as well as many other combinatorial and algebraic objects. In 2006, Ito, Terwilliger, and Weng gave a presentation for $U_q({\mathfrak{sl}}_2)$ in generators $x,y,y^{-1},z$, called the {\it equitable presentation}, and...

10. Graph-restrictive permutation groups and the PSV Conjecture

Prof. Michael Giudici (University of Western Australia)

SYGN Invited Talk

A permutation group $L$ of degree $d$ is called graph-restrictive if there is a constant $c$ such that for every connected graph $\mathrm{\Gamma }$ of valency $d$ admitting a group of automorphisms $G$ with local action $G_v^{\mathrm{\Gamma }(v)}\cong L$ we have that $|G_v|\le c$. Using this terminology, the Weiss Conjecture asserts that every primitive group is graph-restrictive. Poto\v{c}nik, Spiga and Verret...

19. Irreducibility of configurations

Dr Klara Stokes (Universitat Oberta de Catalunya)

In a paper from 1886, Martinetti enumerated small v_3-configurations using a construction that permits to produce a (v +1)_3-configuration from a v_3-configuration. He called configurations that were not constructible in this way irreducible configurations. According to his definition, the irreducible configurations are Pappus’ configuration and four infinite families of configurations. In...

22. On colourings of projective planes and spaces

Prof. Tamás Szőnyi (Eötvös Loránd University, Budapest, Hungary and MTA-ELTE Geometric and Algebraic Combinatorics Research Group)

SYGN Invited Talk

The chromatic number of a hypergraph is the smallest number of colours needed to colour the points so that no edge is monochromatic. Projective planes of order greater than 2 have chromatic number 2, which simply means that projective planes of order greater than 2 have non-trivial blocking sets. We shall be more interested in the upper chromatic number of projective planes and spaces....

25. On groups all of whose undirected Cayley graphs of bounded valency are integral

Istvan Estelyi (University of Primorska)

A finite group $G$ is called Cayley integral if all undirected Cayley graphs over $G$ are integral, i.e., all eigenvalues of the graphs are integers. The Cayley integral groups have been determined by Klotz and Sander [1] in the abelian case, and by Abdollahi and Jazaeri [2], and independently by Ahmady, Bell and Mohar [3] in the non-abelian case. In this talk we will generalize this...

17. On hamiltonian cycles in Cayley graphs with commutator subgroup of order $pq$

Prof. Dave Witte Morris (University of Lethbridge)

Summerschool Lecture

More than 30 years ago, Erich Durnberger used methods of D.Maru\v{s}i\v{c} to prove that if the commutator subgroup of $G$ has prime order~$p$, then every connected Cayley graph on $G$ has a hamiltonian cycle. Maru\v{s}i\v{c} suggested that it should be possible to replace the prime~$p$ with the product of two distinct primes $p$ and~$q$, but this seems to be a much more difficult problem. We...

24. On the number of words of a given GC-content in some cyclic DNA-codes

Prof. Luis Martínez Fernández (University of the Basque Country, Spain)

SYGN Invited Talk

DNA-codes are codes over an alphabet of four letters, corresponding to the four nucleotides Adenine (A), Thymine (T), Guanine (G) and Cytosine (C), which are often represented using the elements of the field ${\mathbb{F}}_4$ or of the ring $\mathbb{Z}/4\mathbb{Z}$. DNA-codes have found numerous applications in biotechnology; for example, they have been used as molecular bar-codes in chemical libraries, or...

2. Pentavalent symmetric graphs of order twice a prime power

Prof. Yan-Quan Feng (Beijing Jiaotong University)

SYGN Invited Talk

A graph is symmetric if its automorphism group acts transitively on ordered adjacent pairs of vertices of the graph. Let $p$ be a prime and let $n\ge 2$ be a positive integer. In this talk, symmetric $\text{Undefined control sequence mz}$-covers of the dipole $\text{Undefined control sequence Dip}$ (the graph with two vertices connected by five multiple edges) are classified and their full automorphism groups are determined. Among others, connected...

12. Prefix-reversal Gray codes

Mr Alexey Medvedev (Sobolev Institute of Mathematics, Novosibirsk, Russia)

Two scenarios are known to get prefix-reversal Gray codes. The first one was given by S. Zaks in 1984 [BIT, 24, 196-204], and the second one was suggested by A. Williams and J. Sawada in 2013 [Electronic Notes in Discrete Math., 44, 357-362]. In this talk we consider other approaches for obtaining prefix-reversal Gray codes. It was noticed that both constructions are based on the independent...

15. Realizations of {4,4} toroidal maps

Dr Isabel Hubard (IMUNAM)

SYGN Invited Talk

There are five symmetry types of maps on the torus of type {4,4}. After revisiting all families of all five symmetry types, we shall show that if every regular toroid of type {4, 4} admits a realisation on a metric space S, then every toroid of type {4,4} admits a realisation on S.

4. Semisymmetric graphs from algebraic LR Structures: Large vertex stabilizers

Prof. Steve Wilson (Northern Arizona University)

SYGN Invited Talk

LR structures are cycle decompositions of tetravalent graphs having two orbits of edges and satisfying some transitivity and flexibility conditions. We construct tetravalent semisymmetric graphs of girth 4 from them, using the partial line graph. This talk will show some general constructions of LR structures from groups of several kinds. We then show LR structures related to dihedral...

0. Skeletal Polyhedra, Polygonal Complexes, and Nets

Prof. Egon Schulte (Northeastern University)

SYGN Invited Talk

Skeletal polyhedra and polygonal complexes in 3-space are finite, or infinite periodic, geometric edge graphs equipped with additional polyhedra-like structure determined by faces (simply closed planar or skew polygons, zig-zag polygons, or helical polygons). The edge graphs of the infinite polyhedra and complexes are periodic nets. We discuss classification results for skeletal polyhedra and...

16. Skew morphisms of finite groups

Prof. Marston Conder (University of Auckland)

SYGN Invited Talk

A skew morphism of a group is a variant of an automorphism, which arises in the study of regular Cayley maps (regular embeddings of Cayley graphs on surfaces, with the property that the ambient group induces a vertex-regular group of automorphisms of the embedding). More generally, skew morphisms arise in the context of any group expressible as a product $AB$ of subgroups $A$ and $B$ with $B$...

9. The Classification of Minimal non-core-2 2-groups with Almost Maximal Class

Ma Xuesong (Capital Normal University)

Let G be a finite groups with order 2^n. G is called core-2 group if $|H:H_G|\le 2$ for all subgroup $H\le G$. In this paper, we classified the minimal non-core-2 2-groups with almost maximal class. Jointed work with Jiyong Chen, Shaofei Du, Jing Xu and Mingyao Xu.

14. Transitive combinatorial structures constructed from finite groups

Dr Andrea Svob (Department of Mathematics, University of Rijeka)

The main subject of the talk is the construction of transitive combinatorial structures from finite groups. The method will be applied on the construction of 2-designs and strongly regular graphs. The structures will be defined on the conjugacy classes of the maximal and second maximal subgroups under the action of finite groups or their maximal subgroups. Constructed structures and their...

6. Vertex-Transitive Digraphs of Order $p^5$ are Hamiltonian

Dr Jun-Yang Zhang (Minnan Normal University)

It is proved that connected vertex-transitive digraphs of order $p^5$ (where p is a prime) are Hamiltonian, and a connected digraph whose automorphism group contains a finite vertex-transitive subgroup $G$ of prime power order such that $G^{\prime }$ is generated by two elements or elementary abelian is Hamiltonian.