• 数学 •

### 具有两个度数的树

1. 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 出版日期:2023-03-25 发布日期:2023-03-23
• 通讯作者: 詹兴致 E-mail:pq@ecust.edu.cn;zhan@math.ecnu.edu.cn
• 作者简介:

### Two-degree trees

Pu QIAO1(), Xingzhi ZHAN2,*()

1. 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:2021-05-07 Online:2023-03-25 Published:2023-03-23
• Contact: Xingzhi ZHAN E-mail:pq@ecust.edu.cn;zhan@math.ecnu.edu.cn

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.

Key words: two-degree tree, diameter, unique graph