Geometric Spanning Trees Minimizing the Wiener Index

Geometric Minimum Spanning Trees

Let S be a set of n points in < d. We present an algorithm that uses the well-separated pair decomposition and computes the minimum spanning tree of S under any Lp or polyhedral metric. It has an expected running time of O(n logn) for uniform distributions. Experimentalresults show that this approachis practical. Under a variety of input distributions, the resultingimplementation is robust and ...

Minimum Dilation Geometric Spanning Trees

The Minimum Dilation Geometric Spanning Tree Problem (MDGSTP) is N P-hard, which justifies the development of heuristics to it. This paper presents heuristics based on the GRASP metaheuristic paradigm for MDGSTP. The input of this problem is a set of points P = {p1, p2, . . . , pn} in the plane. Let the geometric graph G(P) associated with P be the undirected weighted complete graph of n vertic...

The second geometric-arithmetic index for trees and unicyclic graphs

Let $G$ be a finite and simple graph with edge set $E(G)$. The second geometric-arithmetic index is defined as $GA_2(G)=sum_{uvin E(G)}frac{2sqrt{n_un_v}}{n_u+n_v}$, where $n_u$ denotes the number of vertices in $G$ lying closer to $u$ than to $v$. In this paper we find a sharp upper bound for $GA_2(T)$, where $T$ is tree, in terms of the order and maximum degree o...

Wiener index of random trees

The Wiener Index for Weighted Trees

The Wiener index of a graph is the sum of the distances between all pairs of vertices. In fact, many mathematicians have study the property of the sum of the distances for many years. Then later, we found that these problems have a pivotal position in studying physical properties and chemical properties of chemical molecules and many other fields. Fruitful results have been achieved on the Wien...