22 research outputs found
Extremal Values of Ratios: Distance Problems vs. Subtree Problems in Trees
The authors discovered a dual behaviour of two tree indices, the Wiener index and the number of subtrees, for a number of extremal problems [Discrete Appl. Math. 155 (3) 2006, 374-385; Adv. Appl. Math. 34 (2005), 138-155]. Barefoot, Entringer and Székely [Discrete Appl. Math. 80 (1997), 37-56] determined extremal values of σT(w)/σT(u), σT(w)/σT(v), σ(T)/σT(v), and σ(T)/σT(w), where T is a tree on n vertices, v is in the centroid of the tree T, and u,w are leaves in T.
In this paper we test how far the negative correlation between distances and subtrees go if we look for the extremal values of FT(w)/FT(u), FT(w)/FT(v), F(T)/FT(v), and F(T)/FT(w), where T is a tree on n vertices, v is in the subtree core of the tree T, and u,w are leaves in T-the complete analogue of [Discrete Appl. Math. 80 (1997), 37-56], changing distances to the number of subtrees. We include a number of open problems, shifting the interest towards the number of subtrees in graphs
Enumeration of subtrees of trees
Let be a weighted tree. The weight of a subtree of is defined
as the product of weights of vertices and edges of . We obtain a
linear-time algorithm to count the sum of weights of subtrees of . As
applications, we characterize the tree with the diameter at least , which
has the maximum number of subtrees, and we characterize the tree with the
maximum degree at least , which has the minimum number of subtrees.Comment: 20 pages, 11 figure