数学

具有两个度数的树

  • 乔璞 ,
  • 詹兴致
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  • 1. 华东理工大学 数学学院, 上海 200237
    2. 华东师范大学 数学科学学院, 上海 200241

收稿日期: 2021-05-07

  网络出版日期: 2023-03-23

Two-degree trees

  • Pu QIAO ,
  • Xingzhi ZHAN
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  • 1. School of Mathematics, East China University of Science and Technology, Shanghai 200237, China
    2. School of Mathematical Sciences, East China Normal University, Shanghai 200241, China

Received date: 2021-05-07

  Online published: 2023-03-23

摘要

如果一个图只有两个不同的度数, 这个图就称为二度图. 阶数至少为3的二度树具有度数1和 $d $ , 这里 $d $ 是至少为2的整数, 这样的树称为 $(1,d)$ -树. 给定一个正整数 $n $ , 确定了以下信息: (1)存在一个 $n $ $(1,d) $ -树的可能的 $d $ 的值; (2)存在唯一的 $n $ $(1,d)$ -树的可能的 $d $ 的值; (3) $n $ $(1,d)$ -树的最大可能直径. 这些结果提供了一个新的例子, 表明有时候图的行为是由数论性质决定的.

关键词: 二度树; 直径; 唯一图

本文引用格式

乔璞 , 詹兴致 . 具有两个度数的树[J]. 华东师范大学学报(自然科学版), 2023 , 2023(2) : 1 -4 . DOI: 10.3969/j.issn.1000-5641.2023.02.001

Abstract

A graph is called a two-degree graph if its vertices have only two distinct degrees. A two-degree tree of order at least three have two degrees, $ 1 $ and $ d $ for some $ d\geqslant 2; $ such a tree is called a $ (1,d) $ -tree. Given a positive integer $ n, $ we determine: (1) the possible values of $ d $ such that there exists a $ (1,d) $ -tree of order $ n; $ (2) the values of $ d $ such that there exists a unique $ (1,d) $ -tree of order $ n $ , and (3) the maximum diameter of two-degree trees of order $ n. $ The results provide a new example showing that the behavior of graphs may sometimes be determined by number theoretic properties.

参考文献

1 BONDY J A, MURTY U S R. Graph Theory [M]. New York: Springer, 2008.
2 BLASS A, HARARY F, MILLER Z. Which trees are link graphs?. Journal of Combinatorial Theory (Series B), 1980, 29, 277- 292.
3 BROERSMA H, XIONG L, YOSHIMOTO K. Toughness and hamiltonicity in k-trees . Discrete Math, 2007, 307, 832- 838.
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