讲座1： An algebra for weighted threshold graphs

Threshold graphs are generated from one vertex by repeatedly adding a vertex adjacent to all existing vertices or adding an isolated vertex. In the weighted threshold graph, we add a new vertex in step $i$, which is connected to all existing vertices by an edge of weight $w_i$. In this work, we consider the set ${\cal A}_n$ consisting of all Laplacian matrices of weighted threshold graphs of order $n$. We show that ${\cal A}_n$ forms a commutative algebra and find a common basis of eigenvectors for the matrices in ${\cal A}_n$. It follows that the eigenvalues of each matrix in ${\cal A}_n$ can be represented as a linear transformation of the weights.

The talk is based on joint work with Yingyue Ke and Piet Van Mieghem.

讲座2： Generalized spectral characterizations of Eulerian graphs: revisited

Let $G$ be an Eulerian graph on $n$ vertices with adjacency matrix $A$ and characteristic polynomial $\phi(x)$. We show that when $n$ is even (resp. odd), the square-root of $\phi(x)$ (resp. $x\phi(x)$) is an annihilating polynomial of $A$, over $\mathbb{F}_2$. The result was achieved by applying the Jordan canonical form of $A$ over the algebraic closure $\bar{\mathbb{F}}_2$. Based on this, we show that a family of Eulerian graphs are determined by their generalized spectrum among all Eulerian graphs, which significantly simplifies and strengthens the previous result. This is a joint work with Kunyue Li and Hao Zhang.

讲座3： The ranks of tensors and their applications

This talk will address the ranks of tensors, notions of complexity on tensors that extend the matrix rank each in their own way. The ranks of tensors have been successfully applied to several areas such as (among many more others) communication complexity, circuit complexity, quantum information theory, data compression, machine learning, and network analysis, yet despite that, much of the basic understanding of these ranks is still in its very early stages. We will begin by recalling the origins of some of the rank notions on tensors (such as the tensor rank, the slice rank, the partition rank, the R-rank and the subrank) as well as some of the above applications. Next, we will review some of the major advances in the accelerating development of the basic theory of the ranks of tensors that have been witnessed by the last ten years, and in turn some of the resulting improvements in applications. Thereafter, we will focus specifically on the extension of several basic properties of the rank of matrices to the ranks of tensors: first on which generalisations hold for all tensors, and then on how they may be qualitatively strengthened further for specific classes of tensors. Finally, we will outline directions of research that we believe would make progress on central difficulties behind several basic but open questions regarding these ranks.

讲座4：Cyclic difference families, perfect difference families and their applications

Let $v$ be a positive odd integer. A $(v,k,\lambda)$-perfect difference family (PDF) is a collection $\mathcal F$ of $k$-subsets of $\{0,1,\ldots,v-1\}$ such that the multiset $\bigcup_{F\in\mathcal F}\left\{x-y : x,y\in F, x>y\right\}$ covers each element of $\left\{1,2,\ldots,(v-1)/2\right\}$ exactly $\lambda$ times. Perfect difference families are a special class of perfect systems of difference sets. This talk shows that a $(v,4,\lambda)$-PDF exists if and only if $\lambda(v-1) \equiv 0 \pmod{12}$, $v \geq 13$, and $(v,\lambda) \notin \{(25,1),(37,1)\}$. This result resolves a nearly 50-year-old conjecture posed by Bermond. Perfect difference families find applications in radio astronomy, optical orthogonal codes for optical code-division multiple access systems, geometric orthogonal codes for DNA origami, difference triangle sets, additive sequences of permutations, and graceful graph labelings. Perfect difference families can be also seen as a special kind of cyclic difference families. This talk will also give a survey on the recent progress on constructions for cyclic difference families.

1.Willem Haemers (蒂尔堡大学)

2. Wei Wang (西安交通大学)

3.Thomas Karam (上海交通大学)

4.Tao Feng (北京交通大学)