#### Title

Extremal Values of Ratios: Distance Problems vs. Subtree Problems in Trees II

#### Document Type

Article

#### Publication Date

5-6-2014

#### Publication Title

Discrete Mathematics

#### DOI

10.1016/j.disc.2013.12.027

#### ISSN

0012-365X

#### Abstract

We discovered a dual behavior of two tree indices, the Wiener index and the number of subtrees, for a number of extremal problems (Székely and Wang, 2006, 2005). We introduced the concept of *subtree core*: the *subtree core * of a tree consists of one or two adjacent vertices of a tree that are contained in the largest number of subtrees. Let σ(T)denote the sum of distances between unordered pairs of vertices in a tree T and σ_{T}(v)the sum of distances from a vertex v to all other vertices in T. Barefoot et al. (1997) determined extremal values of σ_{T}(w)/σ_{T}(u), σ_{T}(w)/σ_{T}(v), σ(T)/σ_{T}(v), andσ(T)/σ_{T}(w), where T is a tree on n vertices, v is in the centroid of the tree T, and u,ware leaves in T. Let F(T) denote the number of subtrees of T and F_{T}(v) the number of subtrees containing v in T. In Part I of this paper we tested how far the negative correlation between distances and subtrees go if we look for (and characterize) the extremal values of F_{T}(w)/F_{T}(u), F_{T}(w)/F_{T}(v). In this paper we characterize the extremal values of F(T)/F_{T}(v), and F(T)/F_{T}(w), where T is a tree on n vertices, v is in the subtree core of the tree T, and w is a leaf in T-completing the analogy, changing distances to the number of subtrees.

#### Recommended Citation

Szekely, Laszlo A., Hua Wang.
2014.
"Extremal Values of Ratios: Distance Problems vs. Subtree Problems in Trees II."
*Discrete Mathematics*, 322: 36-47.
doi: 10.1016/j.disc.2013.12.027

https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/332