University of Primorska Faculty of Mathematics, Natural Sciences and Information Technologies
-->
SI | EN

Mathematical Research Seminar - Archive

2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010
1 2 3 4 5 6 7 8 9 10 11 12
Datum in ura / Date and time: 27.2.24
(14:00 -- 15:00)
Predavalnica / Location: FAMNIT-VP1
Predavatelj / Lecturer: Joy Morris (University of Lethbridge, Canada)
Naslov / Title: Detecting (Di)Graphical Regular Representations
Vsebina / Abstract:
Graphical and Digraphical Regular Representations (GRRs and DRRs) are a concrete way to visualise the regular action of a group, using (di)graphs. More precisely, a GRR or DRR on the group $G$ is a (di)graph whose automorphism group is isomorphic to the regular action of $G$ on itself by right-multiplication. 
For a (di)graph to be a DRR or GRR on $G$, it must be a Cayley (di)graph on $G$. Whenever the group $G$ admits an automorphism that fixes the connection set of the Cayley (di)graph setwise, this induces a nontrivial graph automorphism that fixes the identity vertex, which means that the (di)graph is not a DRR or GRR. Checking whether or not there is any group automorphism that fixes a particular connection set can be done very quickly and easily compared with checking whether or not any nontrivial graph automorphism fixes some vertex, so it would be nice to know if there are circumstances under which the simpler test is enough to guarantee whether or not the Cayley graph is a GRR or DRR. I will present a number of results on this question.
 
Join Zoom Meeting

https://upr-si.zoom.us/j/85914318577

Meeting ID: 859 1431 8577

Don't miss out on this opportunity to delve into cutting-edge research and expand your mathematical horizons!

 

 


Datum in ura / Date and time: 19.2.24
(15:00 -- 16:00)
Predavalnica / Location: FAMNIT-VP1
Predavatelj / Lecturer: Marko Orel (University of Primorska)
Naslov / Title: Marko Orel
Vsebina / Abstract:

In this talk, a graph Γ = (V, E) is a finite simple graph, which means that V is a finite set and E is a family of its subsets that have two elements. Given two graphs Γ1 = (V1, E1) and Γ2 = (V2, E2), a map Φ : V1 → V2 is
a homomorphism if {Φ(u), Φ(v)} ∈ E2 whenever {u, v} ∈ E1. A bijective homomorphism is an isomorphism in the case {Φ(u), Φ(v)} ∈ E2 if and only if {u, v} ∈ E1. As usual, if Γ1 = Γ2, then a homomorphism/isomorphism is
an endomorphism/automorphism. A core of a graph Γ is any its subgraph Γ0 such that a) there exists some homomorphism from Γ to Γ0 and b) all endomorphisms of Γ0 are automorphisms.
In graph theory, Γ1 and Γ2 are usually treated as ‘equivalent’ if they are isomorphic, i.e. if there exists some isomorphism between them. Less frequent we meet the notion of homomorphically equivalent graphs Γ1, Γ2,
which means that there exists a graph homomorphism Φ : V1 → V2 and a graph homomorphism Ψ : V2 → V1. Here, the notion of a core appears very naturally because two graphs are homomorphically equivalent if and
only if they have isomorphic cores. Often, it is very difficult to decide if there exists a homomorphism between two graphs. In fact, this problem is related to many graph parameters that are hard to compute, such as the
chromatic/clique/independence number. As a result, the study of cores is challenging. In this talk, I will survey some properties of cores, with an emphasis on graphs that either admit a certain degree of ‘symmetry’ or have
‘nice’ combinatorial properties.

 

Everyone is welcome and encouraged to attend.