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

from 29 June 2014 to 5 July 2014
Europe/Ljubljana timezone
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## Contribution List

Displaying 24 contributions out of 24
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 $\ZZ_d$ for an integer $d\geq 2$, metacyclic abelian groups $\ZZ_p^2$, or nonmetacyclic abelian groups $\ZZ_p^3$ for a prime $p$ (see S.F. Du, D. Maru\v si\v c and A.O. Waller, On 2-ar ... More
Presented by Dr. Wenqin XU
Track: 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 arc-trans ... More
Presented by Prof. Shaofei DU
Track: Summerschool Lecture
A Cayley graph $\Gamma={\rm Cay}(G;S)$ 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 $\Gamma$ respects the partition $\mathcal C$ of the edge set of $\Gamma$ if for every $C \in \mathcal C$, we have $\alpha(C) \in \mathcal C$. I will dis ... More
Presented by Prof. Joy MORRIS
Track: 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 in ... More
Presented by Prof. Paul TERWILLIGER
Track: 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 $\Gamma$ such that {\rm (i)} $\Gamma=XY$ for some cyclic groups $X= \langle x \rangle$ and $Y=\langle y \rangle$ with $X \cap Y = \{ 1_{\Gamma} \}$ and {\rm (ii)} there exists an automorphism of $\Gamma$ which sends $x$ and ... More
Presented by Prof. Young Soo KWON
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 Harada ... More
Presented by Dr. Loredana SIMČIĆ
Track: SYGN Invited Talk
There are sevaral variations of vertex and edge colorings of hypergraphs. A {\em vertex coloring\/} of a hypergraph ${\cal H}=(X,{\cal C})$ is a mapping $\phi$ from $X$ to a set of colors $\{1,2,\ldots ,k\} .$ A {\em strict\/ rainbow-free $k$-coloring\/} is a mapping $\phi :X\to\{1,\dots,k\}$ that uses each of the $k$ colors on at least one vertex such that each edge of ${\cal H}$ has at leas ... More
Presented by Prof. György KISS
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 showe ... More
Presented by Gabriel PRETEL
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 sho ... More
Presented by Alison Gordon LYNCH
Track: 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 $\Gamma$ of valency $d$ admitting a group of automorphisms $G$ with local action $G_v^{\Gamma(v)}\cong L$ we have that $|G_v|\leq c$. Using this terminology, the Weiss Conjecture asserts that every primitive group is graph-restrictive. Poto\v{c}nik, Spiga and Verret hav ... More
Presented by Prof. Michael GIUDICI
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 2005, ... More
Presented by Dr. Klara STOKES
Track: 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. The no ... More
Presented by Prof. Tamás SZŐNYI
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 class o ... More
Presented by Istvan ESTELYI
Track: 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 will ... More
Presented by Prof. Dave WITTE MORRIS
Track: 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 $\Bbb F_4$ or of the ring $\Bbb Z/4\Bbb Z$. DNA-codes have found numerous applications in biotechnology; for example, they have been used as molecular bar-codes in chemical libraries, or for b ... More
Presented by Prof. Luis MARTÍNEZ FERNÁNDEZ
Track: 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 $n\geq 2$ be a positive integer. In this talk, symmetric $\mz_p^n$-covers of the dipole $\Dip_5$ (the graph with two vertices connected by five multiple edges) are classified and their full automorphism groups are determined. Among others, connected penta ... More
Presented by Prof. Yan-Quan FENG
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 s ... More
Presented by Mr. Alexey MEDVEDEV
Track: 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.
Presented by Dr. Isabel HUBARD
Track: 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 groups w ... More
Presented by Prof. Steve WILSON
Track: 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 poly ... More
Presented by Prof. Egon SCHULTE
Track: 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 $B$ cyc ... More
Presented by Prof. Marston CONDER
Let G be a finite groups with order 2^n. G is called core-2 group if $|H:H_{G}|\leq 2$ for all subgroup $H\leq G$. 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.
Presented by Ma XUESONG
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 automor ... More
Presented by Dr. Andrea SVOB
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 $G'$ is generated by two elements or elementary abelian is Hamiltonian.
Presented by Dr. Jun-Yang ZHANG