CanaDAM 2021
En ligne, 25 - 28 mai 2021 canadam.math.ca/2021f
       

Coherent configurations with few fibers - Part I
Org: Alyssa Sankey (University of New Brunswick)
[PDF]

STEFAN GYURKI, Slovak University of Technology
The Paulus-Rozenfeld-Thompson graph on 26 vertices  [PDF]

Strongly regular graphs~(SRGs) correspond to homogeneous coherent configurations of~rank~3. In~finding the smallest feasible parameter~set on~which no vertex-transitive~SRG appears was already interested N.~Biggs, one~of~the fathers of~the Algebraic graph theory. In~fact, the smallest order, on~which this happens, is~26, and~the corresponding parameter set is~(26,10,3,4). This parameter~set is~realized by~10 non-isomorphic graphs and~the most symmetric among them is~called the Paulus-Rozenfeld-Thompson graph $T$, having automorphism group of~order~120 isomorphic to~$A_5\times C_2$, acting on~the vertex set with~two orbits of~lengths~20 and~6.

The~talk will provide a~gentle introduction~to a~recently published comprehensive tutorial focusing~on the~graph~$T$ and putting~it into~the context of~classical combinatorial objects.

(This~work is~joint with Mikhail Klin and Matan Ziv-Av.)

BOHDAN KIVVA, University of Chicago
Robustness of the Johnson scheme under fusion and extension  [PDF]

We show that if a coherent configuration X on n vertices or its fission contains a Johnson scheme $J(s,d)$ as a subconfiguration on $(1-c)n$ vertices for a sufficiently small constant $c>0$ and $s>100d^4$, then $X$ itself is a Johnson scheme.

Our result simplifies the conclusion of the Split-or-Johnson lemma, which is one of the key ingredients of Babai's quasipolynomial-time algorithm for the Graph Isomorphism problem.

Additionally, the result can be seen as a strengthening of a 1972 theorem of Klin and Kaluzhnin that corresponds to the case of $c=0$.

Based on a joint work with László Babai.

MIKHAIL MUZYCHUK, Ben-Gurion University of the Negev
On Jordan schemes  [PDF]

In 2003 Peter Cameron introduced the concept of a {\it Jordan scheme} and asked whether there exist Jordan schemes which are not symmetrisations of coherent configurations ({\it proper} Jordan schemes). In my talk I'll present several constructions of infinite series of proper Jordan schemes and present first developments in the theory of Jordan schemes - a new class of algebraic-combinatorial objects. This is a joint work with M. Klin and S. Reichard.

GRIGORY RYABOV, Novosibirsk State University
Infinite family of nonschurian separable association schemes  [PDF]

It is known that there exist infinite families of coherent configurations which are: $(1)$ schurian and separable; $(2)$ schurian and nonseparable; $(3)$ nonschurian and nonseparable. The following question was asked, in fact, in~\cite{1}.

\medskip

\noindent\textbf{Question.}Whether there exists an infinite family of nonschurian separable coherent configurations?

\medskip \noindent We give an affirmative answer to this question. More precisely, we prove the following theorem.

\medskip

\noindent \textbf{Theorem.} For every prime $p\geq 5$, there exists a nonschurian association scheme of degree $4p^2$ which is separable.

\begin{thebibliography}{99}

\bibitem{1} \emph{G.~Chen, I.~Ponomarenko}, Coherent configurations, Central China Normal University Press, Wuhan (2019).

\end{thebibliography}