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 $ 阶 $(1,d) $ -树的可能的 $d $ 的值; (2)存在唯一的 $n $ 阶 $(1,d)$ -树的可能的 $d $ 的值; (3) $n $ 阶 $(1,d)$ -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.

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.