CanaDAM 2021
On-line, May 25 - 28, 2021

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

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}.


\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.


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


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