讲座1： Euclidean designs from spherical embedding of Q-polynomial coherent configurations

Coherent configurations are the generalization of association schemes. It is known that the spherical embedding of Q-polynomial association schemes can form spherical $t$-designs.The concept of Q-polynomial coherent configuration was introduced by Suda in 2022.In this talk, we discuss the spherical embedding of Q-polynomial coherent configuration.We will present a necessary and sufficient condition when the embedding becomes a Euclidean $t$-design (on two concentric spheres).In addition, if we further assume each fiber is a P-polynomial association scheme, then $t\leq 10$.

讲座2：On latin-square graphs avoiding K33

We discuss the problem of existence of latin squares without a substructure consisting of six elements $(r_1, c_2, l_3)$, $(r_2, c_3, l_1)$, $(r_3, c_1, l_2)$, $(r_2, c_1, l_3)$, $(r_3, c_2, l_1)$, $(r_1, c_3, l_2)$, where $(r, c, l)$ means that the cell in the $r$th row and $c$th column contains the symbol $l$. $$ \begin{array}{ccc} \cdot & \mathbf A & \mathbf B \\  \mathbf A  & \cdot  & \mathbf C \\  \mathbf B  & \mathbf C & \cdot \end{array}\qquad $$

Equivalently, the corresponding latin square graph does not have an induced subgraph isomorphic to $K_{3,3}$. The exhaustive search says that there are no such latin squares of order from $3$ to $11$, and there are only two $K_{3,3}$-free latin squares of order $8$, up to equivalence. We repeat the search, establishing also the number of latin $m$-by-$n$ rectangles for each $m$ and $n$ less or equal to $11$. As a switched combination of two orthogonal latin squares of order $8$, we construct a $K_{3,3}$-free (universally noncommutative) latin square of order $16$.

The problem can be generalized to the study of $K_{k+2,k+2}$-free collections of $k$ mutually orthogonal latin squares. For example, among the two linear pairs of orthogonal latin squares over GF$(7)$, one is $K_{4,4}$-free and the other is not.

This is joint work with Aleksandr Krotov.

讲座3：Cubic Pancake graphs

A classical and celebrated result states that two random permutations of a set of $n$ elements almost surely generate either the symmetric or the alternating group of degree $n$, and similar results hold when more generators are considered. In particular, the symmetric group can always be generated by a transposition together with the $n$-cycle, or by any set of $n-1$ transpositions.

Our interest lies in describing sets of three prefix reversals that generate the entire symmetric group. The importance of fixed--degree pancake graphs, in particular, cubic pancake graphs as models of networks was shown. The authors have considered cubic pancake graphs as induced subgraphs of the pancake graph and have identified the combinatorial necessary conditions  for a triple of distinct prefix reversals to generate the symmetric group. Using these necessary conditions, six generating sets were obtained by showing that the set can simulate the three generators of the shuffle-exchange permutation network generated by the right and left shuffles, and the transposition of the first two elements.

We use the following approach to get other generating sets, and to describe in group--theoretical terms necessary and sufficient conditions for these sets to generate the symmetric group. Let $H=\langle r_n, r_m, r_k\rangle$, where $n>m>k$ acts on $\Omega =\{1,\ldots,n\}$. Then we consider different cases on $m$ and $k$ such that $H=\Sn$ with utilizing results from group theory about generating sets of $\Sn$. In particular, all the generating sets for $k=2$ and $k=3$ are characterized. Some other sets are also considered, and structural properties of the corresponding Cayley graphs are discussed. In particular, some results on diameters are shown.

The work is supported by the Mathematical Center in Akademgorodok under the agreement No.~075-15-2025-348 with the Ministry of Science and Higher Education of the Russian Federation.

The talk is based on joint paper with Sa\'ul A.~Blanco, Mikhail P.~Golubyatnikov, Natalia V.~Maslova, and Luka A.~Nikiforov.

讲座4：Eigenfunctions, perfect colorings and related structures for hypergraphs

Perfect colorings (equitable partitions) of graphs have many applications in algebraic combinatorics.  We say that a coloring of vertices of  a hypergraph is \textit{perfect} if any two vertices of the same color have the same set of color ranges of incident hyperedges.

In this talk,  we establish that perfect colorings of hypergraphs have almost the same  algebraic and spectral properties as those for graphs. Moreover, we  show that certain combinatorial structures can be represented as perfect colorings of specific hypergraphs or as eigenfunctions of multidimensional circulant matrices generated by an abelian group.

The talk is based on previous work of the speaker and on recent joint work with Vladimir Potapov.

讲座5：On strong nodal domains for eigenfunctions of Hamming graphs

The Laplacian matrix of the $n$-dimensional hypercube has $n+1$ distinct eigenvalues $2i$, where $0\leq i\leq n$. In 2004, B\i y\i ko\u{g}lu, Hordijk, Leydold, Pisanski and Stadler initiated the study of eigenfunctions of hypercubes with the minimum number of weak and strong nodal domains. In particular, they proved that for every $1\leq i\leq \frac{n}{2}$ there is an eigenfunction of the hypercube with eigenvalue $2i$ that have exactly two strong nodal domains. Based on computational experiments, they conjectured that the result also holds for all $1\leq i\leq n-2$. In this work, we confirm their conjecture for $i\leq \frac{2}{3}(n-\frac{1}{2})$ if $i$ is odd

and for $i\leq \frac{2}{3}(n-1)$ if $i$ is even. We also consider this problem for the Hamming graph $H(n,q)$, $q\geq 3$ (for $q=2$, this graph coincides with the $n$-dimensional hypercube), and obtain even stronger results for all $q\geq 3$.

This is a joint work with Konstantin Vorob'ev.

1.Yan Zhu (上海理工大学)

2. Denis Krotov (索伯列夫数学研究所）

3.Elena Konstantinova (三峡数学研究中心 & 三峡大学 & Akademgorodok 数学中心 & 索伯列夫数学研究所 & 新西伯利亚国立大学)

4.Anna Taranenko (索伯列夫数学研究所)

5.Alexandr Valyuzhenich (河北师范大学)