题目1：New results on Laplacian eigenvalue distribution

主讲人：周波 教授

时间：12月16日8:30-9:30

地点：腾讯会议（899-607-5693）

主办单位：理学院

主讲人简介：

华南师范大学数学科学学院教授、博士生导师，主要兴趣包括组合矩阵论、代数图论及数学化学，近年来主要工作在图与超图谱理论方面, 主持过多项国家自然科学基金项目。

摘要：

Any Laplacian eigenvalue of an n-vertex graph lie in [0,n]. We discuss some new progress on the Laplacian eigenvalue distribution in subintervals of [0,n] that is related with the diameter.

题目2：On determinants of tournaments and $D_k$

主讲人：尤利华 教授

时间：12月16日9:30-10:30

地点：腾讯会议（899-607-5693）

主办单位：理学院

主讲人简介：

尤利华，博士，教授，博士生导师，先后从事组合矩阵论、图谱理论、张量与超图谱理论、极值图论和结构图轮等问题的研究，发表SCI科研论文70余篇，先后主持5项国家自然科学基金委基金项目，3 项广东省自然科学基金委项目和 1 项广州市科信局基金，2011 年入选广州市首批珠江科技新星。

摘要：

The determinant of a tournament $T$ is the determinant of the skew-adjacency of $T$. It is well-known that the determinant of a tournament $T$ with $n$ vertices is $0$ if $n$ is odd, and the square of an odd integer if $n$ is even. For odd $k>0$, the tournament set $\mathcal{D}_k$ consists of tournaments whose all subtournaments have determinant at most $k^2$.

In 2000, Babai and Cameron [L. Babai, P.J. Cameron, Automorphisms and enumeration of switching classes of tournaments, The electronic journal of combinatorics, 7 (1) (2000) R38] proved that a tournament is switching equivalent to a transitive tournament if and only if it contains no diamonds, which implies $T\in \mathcal{D}_1$ if and only if $T$ is switching equivalent to a transitive tournament. In 2023, Boussaïri et al. [A. Boussaïri, S. Ezzahir, S. Lakhlifi, S. Mahzoum, Skew-adjacency matrices of tournaments with bounded principal minors, Discrete Mathematics, 346 (10) (2023) 113552] characterized $\mathcal{D}_3$ as follows: $T\in \mathcal{D}_3$ if and only if $T$ is switching equivalent to a transitive tournament or a transitive blowup of a diamond.

In this talk, we give a construction of tournaments, denoted by $L_n$. Based on this construction, we characterize $\mathcal{D}_5$, obtain some properties of $\mathcal{D}_k$, and present others' characterizations of $\mathcal{D}_1$ and $\mathcal{D}_3$ from a new perspective. Moreover, by using the construction of $L_n$, we prove that there exists a tournament whose determinant is $k^2$ for every odd integer $k$, which solves Question 10 presented in [W. Belkouche, A. Boussaïri, A. Chaïchaâ, S. Lakhlifi, On unimodular tournaments, Linear Algebra and its Applications, 632 (2022) 50-60]. We also prove that $\mathcal{D}_k\backslash \mathcal{D}_{k-2}$ is not an empty set, and there exists a positive integer $m$ such that $\mathcal{D}_k\backslash\mathcal{D}_{k-2}$ contains a tournament with order $n$ for every $n\geq m$, which implies $\mathcal{D}_k\backslash\mathcal{D}_{k-2}$ is an ``infinitely large" set. Furthermore, an open question is proposed for further research and we introduce the new progress on this question.