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运筹学学报（中英文） ›› 2025, Vol. 29 ›› Issue (4): 241-248.doi: 10.15960/j.cnki.issn.1007-6093.2025.04.019

• • 上一篇    

给定直径的超树的第二大无符号拉普拉斯半径

余桂东1,2,†, 袁慧1, 谢欣宇1   

  1. 1. 安庆师范大学数理学院, 安徽安庆 246133;
    2. 合肥幼儿师范高等专科学校初等教育系 (公共教学部), 安徽合肥 230013
  • 收稿日期:2022-06-24 发布日期:2025-12-11
  • 通讯作者: 余桂东 E-mail:guidongy@163.com
  • 基金资助:
    国家自然科学基金(No.11871077),安徽省自然科学基金(No.1808085MA04),安徽省高校自然科学研究重点项目(Nos.KJ2020A0894,KJ2021A0650),安徽省高校研究生科学研究项目(No.KJS20210515),安徽省研究生线下示范课程图论(No.2022xxsfkc038),校级研究生线下课程图论(No.2021aqnuxxkc03)

The second largest signless Laplacian spectral radius of uniform supertree with diameter

YU Guidong1,2,†, YUAN Hui1, XIE Xinyu1   

  1. 1. School of Mathematics and Physics, Anqing Normal University, Anqing 246133, Auhui, China;
    2. Department of Primary Education (Public Teaching Department), Hefei Preschool Education College, Hefei 230013, Auhui, China
  • Received:2022-06-24 Published:2025-12-11

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摘要： 谱极值与极图是当今图谱理论研究的热点问题,学者们非常关注研究图的谱半径达到最大或者最小值所对应的极图。本文刻画了直径为$4$ 的超树的第二大无符号拉普拉斯谱半径的极图。设$\mathbb{S}(m, 4, k)$ 是指有$m$ 条边直径为4 的$k$一致超树的集合, $S_3(m, 4, k)$ 是由直径为4的疏松路$v_1e_1v_2e_2v_3e_3v_4e_4v_5$ 在顶点$v_4$ 处悬挂$m-4$条边得到的$k$ 一致超树。本文首先介绍了超图中边扰动的定义以及相关定理。然后, 根据边扰动等方法,研究得出$S_3(m, 4, k)$ 是$\mathbb{S}(m, 4, k)$中无符号拉普拉斯谱半径达到第二大的图。

关键词: 一致超图, 超树, 无符号拉普拉斯谱半径, 悬挂边

Abstract: The spectral extremal problem and graph are hot issues in the study of graph theory nowadays. Scholars are keen to study the extremal graphs attaining the maximum or minimum spectral radius of graph classes. In this paper, the extremal graph of the second largest unsigned Laplacian spectral radius of a supertree with diameter of $4$ is characterized. Let $\mathbb{S}(m, 4, k)$ be the set of $k$-uniform supertree with $m$ edges and diameter $4$, and $S_3(m, 4, k)$ be the $k$-uniform supertree obtained from a loose path $v_1e_1v_2e_2v_3e_3v_4e_4v_5$ with length $4$ by attaching $m-4$ edges at vertex $v_4$. In this paper, firstly, introducing the definition of edge-shifting operation and related theorems. Then, according to edge-shifting operation, we find $S_3(m, 4, k)$ is the graph with the second largest signless Laplacian spectral radius in $\mathbb{S}(m, 4, k)$.

Key words: uniform hypergraph, supertree, signless Laplacian spectral radius, pendent edge

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