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The Gini index of random trees with an application to caterpillars

Part of: Chemistry Graph theory Combinatorial probability

Published online by Cambridge University Press:  15 September 2017

Hrishikesh Balaji  and
Hosam Mahmoud
Hrishikesh Balaji*
Affiliation:
Winston Churchill High School
Hosam Mahmoud*
Affiliation:
The George Washington University
*
* Postal address: Winston Churchill High School, Potomac, MD 20854, USA. Email address: hrishikeshbalaji@gmail.com
** Postal address: Department of Statistics, The George Washington University, Washington, D.C. 20052, USA. Email address: hosam@gwu.edu
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Abstract

We propose two distance-based topological indices (level index and Gini index) as measures of disparity within a single tree and within tree classes. The level index and the Gini index of a single tree are measures of balance within the tree. On the other hand, the Gini index for a class of random trees can be used as a comparative measure of balance between tree classes. We establish a general expression for the level index of a tree. We compute the Gini index for two random classes of caterpillar trees and see that a random multinomial model of trees with finite height has a countable number of limits in [0, ⅓], whereas a model with independent level numbers fills the spectrum (0, ⅓].

Keywords

Random treechemical treecombinatorial probabilityGini indexWiener indextopological index

MSC classification

Primary: 05C05: Trees 05C12: Distance in graphs 92E10: Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
Secondary: 60C05: Combinatorial probability
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 3 , September 2017 , pp. 701 - 709
Copyright
Copyright © Applied Probability Trust 2017 

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