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活動花絮

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活動資訊

  • 時間：2026/8/17（一）
  • 地點：國立陽明交通大學 理學院 SA307

活動流程

  • 10:00講者：劉家安
    講題：Degree sequence and spectral radius of graphs
    大綱、參考資料

    Let $G$ be a simple graph of order n. The spectral radius $ρ(G)$ of $G$ is the largest eigenvalue of its adjacency matrix. For each positive integer $ℓ$ at most $n$, we give a sharp upper bound for $ρ(G)$ by a function of the first $ℓ$ vertex degrees in $G$, which generalizes a series of previous results. Several applications of these bounds are then provided. The idea of the above result also applies to bipartite graphs. We proved a conjecture stating that the maximum spectral radius of a nearly complete bipartite graph is attained when the missing edges are all incident on a common vertex. For arbitrary bipartite graphs, Bhattacharya, Friedland, and Peled gave the BFP conjecture in 2008. However, we provided counterexamples to the BFP conjecture in 2022. We modify the BFP conjecture, and have recently been working on showing it. This is a joint work with Yen-Jen Cheng, Feng-Lei Fan, and Chih-wen Weng.

  • 11:00講者：鄭硯仁
    講題：The distance matrix of a graph
    大綱、參考資料

    In 1971, Graham and Pollak proved that for a tree $T$ of order $n$, the determinant of the distance matrix of $T$ depends only on $n$; that is, it is independent of the structure of $T$. This result has been extensively studied and has led to many extensions and related results. In this talk, I will introduce several graph metrics, their corresponding distance matrices, and some related results.

  • 12:00午餐自理

  • 2:00講者：林晉宏
    講題：Laplacian matrices and related problems
    大綱、參考資料

    In this talk, we will give an introduction to classical results of the Laplacian matrix of a graph, including the matrix tree theorem, the algebraic connectivity, and the characteristic set. Afterward, we discuss their recent developments and related problems.

  • 3:00講者：梁順維
    講題：圖的絕對重心
    大綱、參考資料

    Miroslav Fiedler 在 1990 年提出的論文 <Absolute Algebraic Connectivity of Trees> 中為了決定樹的絕對代數連通度提出了圖的絕對重心(absolute center of gravity of graph)，並且將其應用於樹上 。 在這次報告中將介紹絕對重心的定義與在樹上如何決定其位置的相關定理；在樹上除了絕對重心外還有與其相似的樹的重心 (centroid of tree) 演算法也會一併介紹，並且比較兩種方法所得出的重心位置。

  • 3:30講者：徐振翔
    講題：Graph Laplacians and Nodal Domains (節點域)
    大綱、參考資料

    每一張簡單圖都能對應到一個拉普拉斯矩陣；若根據需求賦予邊不同的權重，我們將得到一個『一般化拉普拉斯矩陣』。透過求解該矩陣的特徵向量，我們能依據數值的正、負與零點，將原圖分割成數個獨立的區塊（即節點域）。本次報告將深入探討這類分割數量的理論上限，及其在譜圖理論中的實際應用。

  • 4:00講者：丁逸弘
    講題：Three Proofs of Cayley's Formula and Their Interconnections
    大綱、參考資料

    本報告旨在探討計算完全圖生成樹數量的 Cayley's Formula $T_n = n^{n-2}$，並透過三種證明解析其數學結構與內在聯繫。報告首先以 Prüfer Code 切入，展示如何透過算法建立標號樹與序列之間的雙射；接著介紹停車函數，並將其與根生成樹數量建立連結；最後從圖論的視角，簡單介紹矩陣樹定理，並利用雙射將樹結構映射至一種與停車函數有相似性的函數。最後對比這三種方法，說明 Prüfer Code、停車函數跟生成樹如何互相對應，展現標號樹計數在組合數學中的統一性。

如對活動有任何疑問，歡迎利用 jephianlin [at] gmail [dot] com 與 Jephian Lin 聯絡 :smiley: