一类图的特征值

文章将树做了推广,给出了圈树的定义:把树的度数大于3的若干点用相应点度数一样长的圈替换得到的图为圈树。证明了点赋权树T(权重均为正),权和为W,则存在一个点v∈V(T),使得T-v的所有连通片的权和不大于W/2。以此为基础,证明了n阶圈树D,证明了一定存在{u,v},使D-{u,v},所有的连通片的阶都不大于n/2],最后对圈树的一些特征值阶进行了估计。

The Eigenvalue of Some Graphs

In this paper, Abstract. Let T be a tree with maximum degree at least 4. If we replace some vertex of degree k （ k 〉 3 ） of T with k --cycle, we obtain a graph called circle --trees. With each vertex u of T let there be associated a real positive number w（u） , called its weight . Then T , together with these v.eights on its vertices, is called a weighted tree . The sum of all weights on its vertices is called weight sum . In this paper, we prove that, for a weighted tree T with weight sum W, there is a vertex v of T such W that the weight sum of all components of T-- v is at most W/2 . Furthermore, we show that, for circle -- trees D with n vertices, there exist u and v of D such that the order of each components of D-- {u,v} is no more than n/2 , and then estimate some ranks of Eigenvalues of circle --trees.