#### Title

Greedy Trees, Caterpillars, and Wiener-Type Graph Invariants

#### Document Type

Contribution to Book

#### Publication Date

2012

#### Publication Title

Mathematical Chemistry Monographs: Distance in Molecular Graphs Theory

#### ISBN

978-86-6009-012-8

#### Abstract

The extremal questions of maximizing or minimizing various distance-based graph invariants among trees with a given degree sequence have been vigorously studied. In many cases, the so-called greedy tree and the caterpillar are found to be extremal. In this note, we show a “universal property” of the greedy tree with a given degree sequence, namely that the number of pairs of vertices whose distance is at most k is maximized by the greedy tree for all k. This rather strong assertion immediately implies, and is equivalent to, the minimality of the greedy trees with respect to graph invariants of the form W_{f} (T)= ∑_{ {u,v}⊆V (T)} f(d(u, v)) for any nonnegative, nondecreasing function f. With diﬀerent choices of f, one directly solves the minimization problems of distance-based graph invariants including the classical Wiener index, the Hyper-Wiener index and the generalized Wiener index. We also consider the maximization of some of such invariants among trees with a given degree sequence. These problems turned out to be more complicated. Analogous to the known case of the Wiener index, we show that W_{f} (T) is maximized by a caterpillar for any increasing and convex function f. This result also leads to a partial characterization of the structure of the extremal caterpillars. Through a similar approach, the maximization problem of the terminal Wiener index is also addressed.

#### Recommended Citation

Schmuck, Nina S., Stephen G. Wagner, Hua Wang.
2012.
"Greedy Trees, Caterpillars, and Wiener-Type Graph Invariants."
*Mathematical Chemistry Monographs: Distance in Molecular Graphs Theory*, Ivan Gutman and Boris Furtula (Ed.), 12: 195-214: University of Kragujevac.
isbn: 978-86-6009-012-8

https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/198