具有两个度数的树

Two-degree trees

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)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 ≥ 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.