Prof.
Robin Wilson
(The Open University, UK)

Much of this meeting centres around Cayley graphs.
But who was Cayley? What contributions did he make to combinatorics? And what were his connections with graphs and groups?

Joy Morris
(University of Lethbridge, Canada)

A major topic of research in symmetries of graphs is that of finding graphs that admit specific sorts of automorphisms, but do not admit other automorphisms. The study of regular representations is an example of this. Another way to approach this topic is to ask what constraints we can impose on a highly symmetric graph if we want to limit the automorphisms. It is with this approach in mind...

Prof.
Robert Jajcay
(Comenius University, Bratislava, and University of Primorska, Koper)

If we denote by $n(k, d)$ the order of the largest undirected graphs of
maximum degree $k$ and diameter $d$, and let $M(k,d)$ denote the corresponding
Moore bound, then $n(k,d) \leq M(k,d)$, for all $ k \geq 3, d \geq 2 $. While the inequality has been proven strict for all but very few pairs $k$ and $d$, the exact relation between the values $n(k,d)$ and $M(k,d)$ is unknown, and the...

Dr
Boris Zgrablic
(University of Primorska)

The Cayley graph relative to a union of conjugacy classes of a finite group will be discussed: properties, results and open questions.

Gábor Gévay
(University of Szeged, Hungary)

A geometric configuration of type (p_q,n_k) is an incidence structure consisting of
p points and n "blocks" such that each point is incident with q blocks and each
block is incident with k points. The "blocks" can be different geometric figures
such as lines, circles, conics, etc. The first known geometric configurations
originate from classical incidence theorems such as the theorems of...

Dr
Simona Bonvicini
(University of Modena and Reggio Emilia)

Let χ'(G) denote the chromatic index of G and denote by G-e the graph obtained by removing an edge e from G. The graph G is said to be Δ-critical if χ'(G)=Δ+1 and
χ'(G-e)=Δ for any edge e in E(G). We are interested in critical graphs of even order.
It has been proved that there are no critical graphs of even order n≤ 10 and that there are no 3-critical graphs of order 12 and 14 (see [4])....

Dr
Julian Dimitrov
(Employer: University of Mining and Geology, Sofia, Bulgaria; Position: Chief assistant professor)

There is discussed the recovery conditions of continuous mapping from a discrete representation. The mapping is presented in a network of model elements located on a differentiable manifold.
Motivation for this material: a.) The applied models in geomechanics are characterized by large errors of the parameters and uncertainty in results. However, these models are presented by continuous...

Dr
Oznur Yasar Diner
(Kadir Has University)

Edge searching is a combinatorial game where a team of searchers are following a strategy to capture a hidden intruder. It is played on the edges and vertices of a graph. When a fixed number of searchers are available, there is a finite number of forbidden minors, exclusion of which will guarantee that the graph will be intruder free.
In this talk, we give the set of forbidden minors for...

Monika Pilsniak
(AGH University, Krakow, Poland)

\titleoftalk{Graphs with small distinguishing index}
\speaker{{Monika} {Pil\'sniak}}
\university{Department of~Discrete Mathematics\\AGH University, Krakow, Poland}
\email{pilsniak@agh.edu.pl}
\
\begin{abstract}The distinguishing index of~a graph $G$, denoted by $D'(G)$, is
the~least number of~colours in a general edge colouring of~$G$ not
preserved by any non-trivial...

Prof.
Darryn Bryant
(University of Queensland)

I will talk about Hamilton decompositions of graphs, focussing on a number of results from the last few years. In particular I will discuss problems relating to Hamilton decompositions of vertex-transitive graphs and line graphs.

Prof.
Cun-Quan Zhang
(West Virginia University)

This is a survey talk about Tutte's integer flows in Cayley graphs. Alspach conjectured that every connected Cayley graph contains a Hamilton cycle. After almost five decades, Alspach's conjecture remains widely open. Note that every Hamiltonian graph admits a nowhere-zero $4$-flow. The following is a weaker version of Alspach's conjecture (by Alspach, Liu and Z) that every Cayley graph...

Dr
Russ Woodroofe
(University of Primorska)

Two elements $x,y$ *invariantly generate* a group $G$ if any conjugate $x$ together with any conjugate of $y$ generates $G$. Invariant generation of a group $G$ by prime or prime-power elements has consequences for fixed-point-free actions on certain geometries with $G$ actions.
In previous work, John Shareshian and I have shown that, assuming the Riemann hypothesis, the alternating groups...

Prof.
Marston Conder
(University of Auckland)

In this talk I'll give a partial (but possibly complete) answer
to a question asked by Pierre-Emmanuel Caprace at the Groups St Andrews
conference at Birmingham (UK) in August 2017, and investigated at
the Tutte Centenary Retreat in Australia in November 2017.
Caprace asked if there exists a 2-transitive permutation group $P$
such that only finitely many simple groups act...

Gabriel Jakóbczak
(Jagiellonian University)

A vertex coloring of graph satisfies the *majority rule*, if for each vertex $v$ at most half of its neighbors receive the same color as $v$. A coloring which satisfies the majority rule is called *majority coloring*. The problem of such colorings was introduced in [1,5] and continued with different variants in [2,4]. We consider its game version. For given graph $G$ and color set $C$ two...

Prof.
Ismail Naci Cangul
(Uludag University)

Delen and Cangul recently defined a new graph invariant in terms of a degree sequence. By means of this invariant, it is possible to classify the realizations of the given degree sequence. Also this new invariant gives a lot of information about the connectedness, the number of components, loops, pendant edges, chords and multiple edges of the realizations. We shall give some new combinatorial...

Prof.
Donovan Hare
(University of British Columbia)

The cycle space of a graph $G$ is the subspace of the edge space of $G$ over
the 2-element field that is
spanned by the cycles of $G$ (considered as edge-sets of $G$).
Bondy and Lov\'{a}sz (1981) showed that
the cycles through any set of $s-1$ vertices in an $s$-connected graph
generate its cycle space. But what if these cycles are restricted to be of odd length?
In this talk, we...

Prof.
Saad El-Zanati
(Illinois State University)

A *loose $m$-cycle* is a 3-uniform hypergraph with vertex set $\{v_1, v_2, \ldots, v_{2m}\}$ edge set $\{\{v_1,v_2,v_3\}, \{v_3,v_4,v_5\}, \ldots, \{v_{2m-1},v_{2m},v_1\}\}$. We consider the problem of decomposing $K_{v}^{(3)}$, the complete 3-uniform hypergraph of order $v$, into edge-disjoint loose $m$-cycles. We settle the problem in the case $m=4$.

Dr
Ademir Hujdurović
(University of Primorska)

One of the central questions in the study of graphs admitting a certain degree of symmetry is determining how large their automorphism groups can be. For graphs of fixed valency, this is equivalent with determining possible orders of vertex-stabilizers. The famous Tutte's result from 1948 implies that vertex stabilizer of a cubic arc-transitive graph can have order at most 48. Arc-transitive...

Aleksander Malnic
(University of Ljubljana & University of Primorska)

A polynomial $f(x) = a_0 + a_1 x + \ldots + a_n x^{n}$ over the prime field $\mathbb{Z}_p$, where $p$ is odd, is reflexible if there exists $\lambda \in \mathbb{Z}_p^*$ such that $\lambda a_{n-i} = a_i$ (type 1) or $\lambda a_{n-i} = (-1)^i a_i$ (type 2), for all indices $i = 0, 1, \ldots, n$. Such polynomials were instrumental in the classification of quartic arc transitive graphs arising...

Dr
Modjtaba Ghorban
(Department of Mathematics, Shahid Rajaee Teacher Training University)

In authors proposed a formula for computing the spectrum of Cayley graph $\Gamma=Cay(G,S)$ with respect to the character table of $G$ where $S$ is a symmetric normal subset of $G$.
Let $q$ be a power of prime number $p$. A representation of degree $n$ of group $G$ is a homomorphism $\alpha: G \to GL(n,q)$, where $\alpha(g)=[g]_{\beta}$ for some basis $\beta$.
A character table is a...

Prof.
Mohammad A. Iranmanesh
(Yazd University)

**Abstract**
Let $G$ be a group and $S$ be a subset of $G$. Then $BCay(G,S)$, bi-Cayley graph of $G$ with respect to $S$, is an undirected graph with vertex-set $G\times\{1,2\}$ and edge-set $\{\{(g,1),(sg,2)\}\mid g\in G, s\in S\}$. For $\sigma\in Aut(G)$ and $g\in G$, we have $BCay(G,S)\cong BCay(G,gS^\sigma)$. A bi-Cayley graph $BCay(G,S)$ is called a
$BCI$-graph if for any bi-Cayley...

Dr
Mark Ellingham
(Vanderbilt University, USA)

White, Pisanski and others have proved a number of results on the existence of quadrilateral embeddings of cartesian products of graphs; in some cases these provide minimum genus embeddings. In a 1992 paper Pisanski posed three questions. First, if $G$ and $H$ are connected $1$-factorable $r$-regular graphs with $r \ge 2$, does the cartesian product $G \times H$ have an orientable...

Dr
Adnan Melekoglu
(Adnan Menderes University)

It is known that every Riemann surface of genus $g$ can be expressed in the form
$\mathbb{U}/\Omega$, where $\mathbb{U}$ is the Riemann sphere $\Sigma$,
the Euclidean plane $\mathbb{C}$, or the hyperbolic plane $\mathbb{H}$,
depending on whether $g$ is $0$, $1$ or $>1$, respectively, and $\Omega$
is a discrete group of isometries of $\mathbb{U}$.
A Riemann surface...

Danny Dyer
(Memorial University of Newfoundland)

Broadly, there are two classical pursuit-evasion problems in graphs. One is cops and robbers, where cops move along vertices in pursuit of a robber that is slow, visible, and also moving on vertices. The other is edge-searching, or sweeping, where agents pursue a fast, invisible evader who moves on edges and vertices. This talk will talk about some of the history of these two problems, as they...

Prof.
Gabor Korchmaros
(University of Basilicata)

A {\em unital} is defined to be a set of $q^3 + 1$ points equipped with a family of subsets, each of size $q + 1$, such that every pair of distinct points are contained in exactly one subset of the family. Such subsets are usually called {\em blocks} so that unitals are block-designs $2-(q^{3}+1,q+1,1)$. Unitals are known to play an important role in many investigations in finite geometry. ...

Dr
Nina Mostarac
(Department of Mathematics, University of Rijeka)

In this talk we will give constructions of self-orthogonal and self-dual codes, with respect to certain scalar products, with the help of orbit matrices of block designs and quotient matrices of symmetric (group) divisible designs (SGDDs) with the dual property. First we will describe constructions from block designs and their extended orbit matrices, where the orbit matrices are induced by...

Dr
Marija Maksimović
(Department of Mathematics University of Rijeka)

In this talk we will show that under certain conditions submatrices of orbit matrices of strongly regular graphs span self-orthogonal codes. We apply this method to construct self-orthogonal binary linear codes from column orbit matrices of the triangular graph $T(2k)$ with at most 120 vertices.
Moreover, we construct linear codes from row orbit matrices of strongly regular graph with...

Dr
Emanuele Brugnoli
(University of Palermo)

A $2-(v,k,1)$ design or, also, a Steiner $2$-design is said to be cyclic if it admits an automorphism cyclically permuting all its points. To establish the number NC$(v,k)$ of cyclic $2-(v,k,1)$ designs is in general not feasible and very little is known about this number. By ``playing'' with $(v,k,1)$ difference families, some lower bounds on NC$(v,k)$ are given. In particular, for primes...

Mr
Primož Potočnik
(University of Ljubljana)

A few topics regarding symmetries of finite graphs that I find interesting, intriguing and worth studying shall be presented.

Prof.
Marco Buratti
(Universita di Perugia)

I will survey known results and open problems about the automorphism groups of Steiner triple systems and Kirkman triple systems.

Dr
Primož Šparl
(University of Ljubljana and University of Primorska)

In the generalized graph truncation construction, one replaces each vertex of a $k$-regular graph $\Gamma$ with a copy of a graph $\Upsilon$ of order $k$.
In this talk, which is based on joint work with Eduard Eiben and Robert Jajcay, we focus on symmetry properties of generalized truncations, especially in connection
to the symmetry properties of the graphs $\Gamma$ and $\Upsilon$ used in...

Prof.
Adrian Pastine
(Universidad Nacional de San Luis)

The Hamilton-Waterloo problem asks for a decomposition of the complete graph into $r$ copies of a 2-factor $F_{1}$ and $s$ copies of a 2-factor $F_{2}$ such that $r+s=\left\lfloor\frac{v-1}{2}\right\rfloor$. If $F_{1}$ consists of $m$-cycles and $F_{2}$ consists of $n$ cycles, then we call such a decomposition a $(m,n)-$HWP$(v;r,s)$. The goal is to find a decomposition for every possible pair...

Dr
Sara Sabrina Zemljic
(Comenius University Bratislava)

A $(k,g)$-graph is a simple undirected $k$-regular graph with girth $g$. A $(k,g)$-cage is a $(k,g)$-graph with the least possible number of vertices. Cages have been studied quite intensively yet for most pairs $(k,g)$ the corresponding cages are still unknown.
There is an obvious lower bound on the order of a $(k,g)$-cage, called the Moore bound (i.e., the number of vertices needed for...