讲座1： Attempts on Haemers' conjecture

Haemers conjectures that almost all graphs are determined by their spectra, in other words almost all graphs have no cospectral mate or ``one can hear the shape of almost all graphs''. Suppose $G \sim \mathcal{G}(n, p)$ is a random graph with each edge chosen independently with probability $p$ with $0 < p < 1$. Then $$\Pr(G \text{ is not controllable}) + \sum_{\ell = 2}^{n^{n^2}} \Pr(G \text{ has a generalized copsectral mate with level } \ell) \to 0$$ as $n \to \infty$ implies that almost all graphs are determined by their generalized spectra. It is known that almost all graphs are controllable. We show that almost all graphs have no cospectral mate with fixed level $\ell$, namely $$\Pr(G \text{ has a copsectral mate with level } \ell) \to 0$$ as $n \to \infty$ for every $\ell \geq 2$. The result can also be interpreted in the framework of random symmetric integral matrices.

This talk is based on joint work with Wei Wang.

讲座2： Creating divisible design graphs using affine designs

A $k$-regular graph with $v$ vertices is called a divisible design graph if its vertex set can be partitioned into $m$ classes of size $n$, and any two distinct vertices in the same class have $\lambda_1$ common neighbours, and any two vertices from different classes have $\lambda_2$ common neighbours. Recently,  the speaker presented several new constructions that generate infinite sequences of divisible design graphs. These constructions develops ideas of W.D.~Wallis, D.G.~Fon-Der-Flaass, and M.~Muzychuk. Some of the divisible design graphs from presented by the speaker  were used to construct the new strongly regular graphs with the parameters of the complement symplectic graphs. Possible divisible design graphs suitable for constructing strongly regular graphs were studied by the speaker in two joint works with A. Gavrilyuk.  In the talk we discourse old and introduce a new prolific constructions of infinite families of divisible design graphs with new parameters.

We use a symmetric $2$-design with parameters $(m,\kappa,\lambda)$ and $m$ affine designs $AR(q,r)$. Let $q$ be a power of $2$ and let $\kappa = (q^2 r - 1)/(q - 1)$ be the number of parallel classes of blocks in affine designs. then there are divisible design graphs with parameters $v = q^2 r m, \quad k = q r (q^2 r - 1)/(q - 1),$   $\lambda_1 = q r (q r -1)/(q-1),   \quad \lambda_2 = \lambda r,$ $ m,  \quad n=q^2 r,$ and $v = q^3 r (q r - 1)/(q - 1),\quad k = q^2 r (q r - 1)/(q - 1),$$\lambda_1 = q r (q r -q)/(q-1),\quad \lambda_2 =  q r (q r - 1)/(q - 1),$$m = q^2(q r - 1)/(q-1),\quad n = q r$.

讲座3：Divisible design graphs with selfloops

A $k$-regular graph $\Gamma$ on $v$ vertices is called a divisible design graph (DDG for short) with parameters $(v,k,\lambda_1,\lambda_2,m,n)$ if the vertex set $V$ of $\Gamma$ can be partitioned into $m$ classes each of size $n$ such that any two distinct vertices of the same class have precisely $\lambda_1$ common neighbours and any two distinct vertices of different classes have precisely $\lambda_2$ common neighbours. DDG's were introduced as a bridge between (group) divisible designs and graphs, and the graphs under consideration were undirected and without loops. In this work, we introduce a natural extension of DDGs to include possible selfloops (LDDG's for short) and develop a basic theory for the same. We describe two infinite families of such graphs, some members of which are (undirected) DDG's. We state some theoretical results including the spectrum of such LDDG's. We also classify all examples satisfying parameter restrictions. Finally, we discuss a procedure called dual Seidel switching, which constructs new LDDGs from others. This is based on joint work with Bart De Bruyn and Sergey Goryainov.

讲座4：A new infinite family of divisible design graphs related to antipodal distance-regular graphs of diameter 3

We consider only simple graphs. A \emph{divisible design graph} with parameters $(v,k,\lambda_1,\lambda_2,m,n)$ is a $k$-regular graph on $v$ vertices such that its vertex set can be partitioned into $m$ classes of size $n$ where any two distinct vertices from the same class have exactly $\lambda_1$ common neighbours and any two vertices from different classes have exactly $\lambda_2$ common neighbours. Divisible design graphs were introduced as a bridge between graph theory and the theory of group divisible designs. Since then, tens of constructions of divisible design graphs have been introduced.

In the monograph on distance-regular graphs published in 1989, a construction of antipodal distance-regular graphs of diameter 3 was given. This construction uses a vector space of an even dimension equipped with a nondegenerate symplectic bilinear form. The construction admits the vector spaces over all finite fields $\mathbb{F}_q$. In our work we show that in the case when $q$ is even it is possible to slightly modify this construction by plugging a difference set in $\mathbb{F}_q^+$ into it. This leads to a new infinite family of divisible design graphs. In the talk we discuss this modification and connection of the resulting graphs with the parabolic affine polar graphs.

The authors of \cite{BCN89} further discuss a few generalisations of the construction of distance-regular graphs given in \cite[Proposition 12.5.1]{BCN89}. In our work we show that one of these generalisations, which depends on a subgroup $A$ in $\mathbb{F}_q^+$, can be modified by plugging a difference set in the quotient group of $\mathbb{F}_q^+/A$ into it. This leads to a wider infinite family of divisible design graphs, which includes the family of divisible design graphs mentioned above. The main goal of this talk is to discuss this wider family of divisible design graphs.

This is joint work with Bart De Bruyn, Sergey Goryainov and Ruihan Xie.

讲座5：New infinite families of divisible design graphs, which are covers of strongly regular polar graphs

We consider only simple graphs. A \emph{divisible design graph} with parameters $(v,k,\lambda_1,\lambda_2,m,n)$ is a $k$-regular graph on $v$ vertices such that its vertex set can be partitioned into $m$ classes of size $n$ where any two distinct vertices from the same class have exactly $\lambda_1$ common neighbours and any two vertices from different classes have exactly $\lambda_2$ common neighbours. Divisible design graphs were introduced as a bridge between graph theory and the theory of group divisible designs. Since then, tens of constructions of divisible design graphs have been introduced.

In the monograph on distance-regular graphs published in 1989, a construction of antipodal distance-regular graphs of diameter 3 was given. This construction uses a vector space of dimension 2 equipped with a nondegenerate symplectic bilinear form. Note that the construction admits the vector spaces over all finite fields $\mathbb{F}_q$. Another ingredient of the construction above is a subgroup $N$ of index $r \ge 2$ in $\mathbb{F}_{q}^*$. In our work we show that it is possible to slightly modify this construction by plugging a difference set in the (cyclic) quotient group $\mathbb{F}_q^+/N$ into it and letting the dimension of the vector space be an arbitrary positive even integer. This leads to a new infinite family of divisible design graphs that are $r$-covers of the corresponding symplectic strongly regular polar graphs.

Further, we mimic the construction of divisible design graphs above by replacing the symplectic bilinear form with a bilinear (resp. sesquilinear) form obtained by the polarisation of  hyperbolic, elliptic and parabolic quadratic forms (resp. by the polarisation of the Hermitian form). This gives a few more infinite families of $r$-covers of strongly regular polar graphs (see \cite[Section 2.6, Section 2.7]{BV22}). These infinite families of $r$-covers contain infinite subfamilies of non-trivial divisible design graphs in the elliptic, parabolic and Hermitian cases.

This is joint work with Bart De Bruyn and Sergey Goryainov.

1. Da Zhao (华东理工大学)

2. Vladislav Kabanov (河北师范大学)

3. Anwita Bhowmik (河北师范大学)

4. Ruilin Ma (河北师范大学)

5. Weihao Yan (河北师范大学)