Time: 09:30 - 10:00
Speaker: Dragan Stevanovic
Affiliation: Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade, Serbia
Title: On the numbers of walks in graphs
Abstract: It is well known that the entries of the powers of adjacency matrix represent the numbers of walks between appropriate vertices in a graph, which allows one to conclude that if one graph $G$ has more walks than another graph $H$ for infinitely many different walk lengths, then the spectral radius of $G$ is at least as large as the spectral radius of $H$. In this lecture we will showcase a few interesting ways to compare numbers of walks among graphs, that combine injective embeddings, common factors in characteristic polynomials and counting special types of walks. These approaches recently led to the resolution of Belardo-Li Marzi-Simi\'c conjectured extension of the classical Li-Feng lemma, to the proof that the ordering of starlike trees with constant number of vertices coincides with the shortlex order of sorted sequences of their branch lengths, and to the upper and lower cubic bounds on the spectral radius of threshold graphs.
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Time: 10:10 - 10:40
Speaker: Alireza Abdollahi
Affiliation: University of Isfahan, Isfahan, Iran
Title: Some problems about non-commuting graph of nilpotent and solvable groups
Abstract: Let $G$ be a non-abelian group. The non-commuting graph of $G$, denoted by $\Gamma_G$, is the graph whose vertex set is $G$ and two vertices are adjacent if they do not commute. We review some open problems and solved problems about non-commuting graphs of finite groups.
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Time: 11:10 - 11:40
Speaker: Filiz Yildiz
Affiliation: Hacettepe University, Ankara, Turkey
Title: A directed graph associated with a $T_0$-quasi-metric space
Abstract: It is well known that there are standard connections between the theory of graphs and the theory of metric spaces. Similarly, there exist well-known and less well-known relations between the directed (as well as undirected) graphs and $T_0$-quasi-metric spaces. We survey some of these results and add a few new observations, which complement and refine earlier works in the field in which the symmetry graph of a $T_0$-quasi-metric space was studied. This is a joint work with Prof. H.P. K\"unzi.
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Time: 11:50 - 12:10
Speaker: Maurizio Brunetti
Affiliation: University of Naples Federico II, Naples, Italy
Title: The Hoffman Program for the $A_{\alpha}$-matrix
Abstract: The Hoffman program with respect to any real or complex square matrix $M$ associated to a graph $G$ stems from A. J. Hoffman’s pioneering work on the limit points for the spectral radius of adjacency matrices of graphs less than $\sqrt{2 + \sqrt{5}}$. The program consists of two aspects: finding all the possible limit points of $M$-spectral radii of graphs and detecting all the connected graphs whose $M$-spectral radius does not exceed a fixed limit point. In this paper, we summarize the results on this topic concerning the adjacency, the Laplacian and the signless Laplacian matrix, and present the new achievements related to the Nikiforov's $A_{\alpha}$-matrix.
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Time: 14:00 - 14:30
Speaker: Enric Ventura Capell
Affiliation: Universitat Politècnica de Catalunya, Barcelona, Catalonia
Title: The degree of commutativity/nilpotency of an infinite group
Abstract: There is a classical result saying that, in a finite group, the probability that two elements commute is never between $5/8$ and $1$ (i.e., if it is bigger than $5/8$ then the group is abelian). The are versions of this result for compact groups working with appropriate measures. We make an adaptation of this notion for finitely generated infinite discrete groups (w. r. t. a fixed finite set of generators) as the limit of such probabilities, when counted over successively growing balls in the group. This asymptotic notion is a lot more vague than in the finite setting, but we are still able to prove some interesting results:
(1) with some hypothesis the limit exists and is independent from the set of generators;
(2) a Gromov-like result: “for any finitely generated, residually finite group $G$ of subexponential growth, the commuting degree of $G$ is positive if and only if $G$ is virtually abelian“.
I will also mention some generalizations in two directions: changing to other distributions (i.e., other directions to infinity) and other equations (degree of $r$-nilpotency).
This is joint work with Y. Antolin, A. Martino, M. Tointon, and M. Valiunas.
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Time: 14:40 - 15:10
Speaker: Stephan Wagner
Affiliation: Stellenbosch University, Stellenbosch, South Africa
Uppsala University, Uppsala, Sweden
Title: The mean subtree order
Abstract: The study of the mean subtree order, i.e., the average number of vertices in a subtree of a given tree, goes back to the work of Jamison in the 1980s. Among other things, Jamison proved that the path $P_n$ has the minimum mean subtree order among all trees with $n$ vertices. The corresponding maximisation problem is more complicated and was left as an open question. This is just one of many interesting problems and conjectures that can be found in Jamison's papers. Some of them were only resolved quite recently, and some are still open. In my talk, I will give a survey of the recent progress on these questions and mention some directions for future research.
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Time: 15:50 - 16:20
Speaker: Peter Dankelmann
Affiliation: University of Johannesburg, Johannesburg, South Africa
Title: On distances in graphs embeddable in surfaces
Abstract: In this talk we consider finite planar graphs, i.e., graphs that can be embedded in the plane such that no two edges cross. We present several old and new results on distance measures in such graphs, such as the diameter, the oriented diameter, the Wiener index and eccentricities of vertices.
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Time: 16:30 - 16:50
Speaker: Bettina Wilkens
Affiliation: University of Namibia , Windhoek, Namibia
Title: Subadditive mappings and just coverable groups
Abstract: Let $G$ be a group and $S$ a set. A map $f : G \to \mathcal{P}(S)$ is called subadditive if $f (xy) \subseteq f (x) \cup f (y)$ for all $x, y \in G$. Let $T_f = \bigcup_{x\in G}f(x)$. Placing an upper bound $n$ on $|f (x)|$ for $x \in G$ will result in $|T | ≤ 2n$; we prove this. Afterwards, we turn to describe finite groups in which $|T_f | ≤ \max_{x \in G} |f (x)| + 1$ holds for all subadditive maps; these groups will be named ”just coverable”. We have the following Theorem: A finite-$p$ group is just coverable if and only if it has a uniserially embedded cyclic subgroup.
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Time: 17:00 - 17:25
Speaker: Nikola Koceic-Bilan
Affiliation: University of Split, Split, Croatia
Title: Algebraic topology invariants in the coarse shape
Abstract: Developing of the coarse shape theory, which functorially generalizes shape theory, was followed by introducing corresponding algebraic invariants. In this talk we will present some aspects of contributions of the coarse shape theory to algebraic topology. We will discuss some interesting properties of (topological) coarse shape groups, coarse shape pro-groups, pro*-groups and coarse shape homology groups. Especially, we will compare these algebraic invariants with the analogues algebraic objects in the shape and homotopy theory emphasizing their advantages in studying and classifying locally bad topological spaces.