Seminár z algebraickej teórie grafov - Vladislav Kabanov (30.4.2021)
v piatok 30.4.2021 o 13:00 hod. online formou
Od: Martin Mačaj
Prednášajúci: Vladislav Kabanov (Krasovskii Institute of Mathematics and Mechanics UB RAS, Yekaterinburg, Russia)
Názov: Deza graphs as a generalization of strongly regular graphs
A Deza graph with parameters (v,k,b,a) is a k-regular graph on v vertices in which the number of common neighbors of any two distinct vertices takes two values b or a, where b more or equals a. The only difference between strongly regular graphs and a Deza graphs is that the number of common neighbors of any two distinct vertices does not necessarily depend on whether they are adjacent or not. In addition to strongly regular graphs this class contains divisible design graphs and some other graphs. A k-regular graph on v vertices is a divisible design graph with parameters (v,k,\lambda_1 ,\lambda_2 ,m,n) when its vertex set can be partitioned into m classes of size n, such that any two distinct vertices from the same class have \lambda_1 common neighbors, and any two vertices from different classes have \lambda_2 common neighbors.
In the first part of this talk some general spectral properties of Deza graphs will be done. Due to their spectral properties divisible design graphs are one of the most interesting class of Deza graphs. The second part of the talk will be devote to divisible design Cayley graphs and its constructions. These constructions depends of some special permutations. The problem of isomorphism of such graphs according of the permutations arises, which has not been solved.
Due to recent restrictions imposed on us in response to the worsening of the situation with Covid infections in Slovakia, we will switch to distance mode. You will receive an MS Teams link over which you will be able to follow the presentation, ask questions, comment, and possibly even offer some solutions. While it is not better than being personally in the lecture room, it is not a terrible way to keep in touch either. Let us hope that we will soon be able to attend the meetings in person.