Tree centroid properties clarification

The paper is devoted to the tree centroid properties clarification. Attention of the authors was attracted by the popular problem of (binary) partition of a graph. The solution is known only by brute force algorithm. It was found that for a "economical" partition of a tree it makes sense to consider partitions in the neighborhood of centroid vertices, the definition of which is presented. In the paper, we proposed proofs connected with the limitation of their weight. It is also proved that if there are two centroid vertices in a tree, they are adjacent. In what follows, it is noted that three such vertices can not be in the tree. The corresponding statements are made. According to the first one, any vertex of a tree with a certain restriction on its weight is centroid. According to one of the points of the second statement, if there are two centroid vertices in the tree, the order of the tree is an even number. The third statement says that if a tree has a centroid vertex of limited weight, there is another centroid vertex of the same weight and adjacent to the first one. To prove the propositions, we consider the branch of greatest weight with a centroid vertex and take in this branch another vertex adjacent to the centroid. In this paper, Jordan's theorem is used, three images are used in the presentation of the material.