题目1：Three New Invariants for Cospectral Graphs；
主讲人：王卫 教授
时间：2024年12月10日星期二上午8:30—9:30
地点：理学院会议室308
主办单位：理学院
主讲人简介：
王卫，西安交通大学数学与统计学院教授、博士生导师。主要研究领域为代数图论与组合最优化。在图谱理论的研究中对图的广义谱刻画问题做出了一些原创性的工作，在组合优化领域中对一些NP-困难组合优化问题设计出了一些好的近似算法。在J. Combin. Theory，Ser B, European J. Combin. 以及IEEE/ACMTransactions系列等组合图论刊物上发表研究论文80余篇，主持（完成）国家自然科学基金面上项目四项。目前担任中国运筹学会图论与组合分会常务理事、陕西省工业与应用数学学会理事长及国际刊物“Linear Algebra Appl.” “Discrete Mathematics, Algorithms and Applications”编委等。
摘要：
An invariant for cospectral graphs is a property shared by all graphs. In this talk, we give three new invariants for cospectral graphs, characterized by their arithmetic nature. Based on this, we are able to show that under conditions, every graph cospectral with a graph $G$ is determined by its generalized spectrum. This is a joint work with Yizhe Ji, Quanyu Tang and Hao Zhang.
题目2：Vertex connectivity, edge connectivity, edge-disjoint spanning trees and eigenvalues of (multi-)graphs；
主讲人：王力工 教授
时间：2024年12月10日星期二上午9:30—10:30
地点：理学院会议室308
主办单位：理学院
主讲人简介：
王力工，西北工业大学教授、博士生导师，荷兰Twente大学博士，研究方向为图论及其应用，主要研究包括：图谱理论，有向图与超图的谱性质，整图的刻画，图的Turán数，图的Gallai-Ramsey数等，主持国家自然科学基金多项，在《Journal of Graph Theory》、《Discrete Mathematics》、《Discrete Applied Mathematics》、《Electronic Journal of Combinatorics》、《Linear Algebra and its Applications》等国内外重要学术期刊发表SCI论文140多篇。
摘要：
Let $\lambda_{i}(G)$, $\mu_{i}(G)$ and $q_{i}(G)$ denote the $i$th largest eigenvalue of adjacency matrix $A(G)$, Laplacian matrix $L(G)$ and signless Laplacian matrix $Q(G)$ of a simple graph or a multigraph $G$, respectively. Denote by $\kappa(G)$ vertex connectivity and by $\kappa'(G)$ edge connectivity of $G$, respectively. The maximum number $\tau(G)$ of edge-disjoint spanning trees contained in $G$ is called the spanning tree packing number of $G$. In this talk, we introduce some recent results about the relationships between $\lambda_{i}(G)$, $q_{i}(G)$ or $\mu_{n-i+1}(G)$ and the bounds on $\kappa(G)$, $\kappa'(G)$ and $\tau(G)$ of a simple graph (or a multigraph) $G$ for $i=2,3,4$, respectively. In addition, we mainly present an upper bound for $\lambda_{3}(G)$ in a $d$-regular (multi-)graph $G$ which guarantees that $\kappa(G)\geq t+1$, which is based on the result of Abiad et al. [Spectral bounds for the connectivity of regular graphs with given order, \emph{Electron. J. Linear Algebra} 34:428--443, 2018]. Furthermore, we improve the upper bound for $\lambda_{3}(G)$ in a $d$-regular multigraph which assures that $\kappa(G)\geq 2$. This is a joint work with Tingyan Ma and Yang Hu.
题目3：Anti-Ramsey Number of Friendship Graphs
主讲人：鲁红亮 教授
时间：2024年12月10日星期二上午10:30—11:30
地点：理学院会议室308
主办单位：理学院
主讲人简介：
鲁红亮，西安交通大学数学与统计学院教授，博士生导师；2004年获西北工业大学数学系学士学位，2010年6月获南开大学理学博士学位。主要研究图的度约束因子与超图的匹配问题，解决了多个图因子及匹配研究领域公开问题，已发表论文60余篇，先后主持四项自然科学基金项目。
摘要：
An edge-colored graph is called \textit{rainbow graph} if all the colors on its edges are distinct. For a given positive integer $n$ and a family of graphs $\mathcal{G}$, the anti-Ramsey number $ar(n, \mathcal{G})$ is the smallest number of colors $r$ required to ensure that, no matter how the edges of the complete graph $K_n$ are colored using exactly $r$ colors, there will always be a rainbow copy of some graph $G$ from the family $\mathcal{G}$. A friendship graph $F_k$ is the graph obtained by combining $k$ triangles that share a common vertex. In this paper, we determine the anti-Ramsey number $ar(n, \{F_k\})$ for large values of $n$. Additionally, we also determine the $ar(n, \{K_{1,k}, kK_2\}$, where $K_{1,k}$ is a star graph with $ k+1$ vertices and $kK_2$ is a matching of size $k$.
