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DEGREE-BASED GINI INDEX FOR GRAPHS

Published online by Cambridge University Press:  14 March 2019

Carly Domicolo  and
Hosam Mahmoud
Carly Domicolo
Affiliation:
Department of Statistics, The George Washington University, Washington, D.C. 20052, USA E-mail: carlydomicolo@gwu.edu; hosam@gwu.edu
Hosam Mahmoud
Affiliation:
Department of Statistics, The George Washington University, Washington, D.C. 20052, USA E-mail: carlydomicolo@gwu.edu; hosam@gwu.edu
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Abstract

In Balaji and Mahmoud [1], the authors introduced a distance-based Gini index for rooted trees. In this paper, we introduce a degree-based Gini index (or just simply degree Gini index) for graphs. The latter index is a topological measure on a graph capturing the proximity to regular graphs. When applied across the random members of a class of graphs, we can identify an average measure of regularity for the class. Whence, we can compare the classes of graphs from the vantage point of closeness to regularity.

We develop a simplified computational formula for the degree Gini index and study its extreme values. We show that the degree Gini index falls in the interval [0, 1). The main focus in our study is the degree Gini index for the class of binary trees. Via a left-packing transformation, we show that, for an arbitrary sequence of binary trees, the Gini index has inferior and superior limits in the interval [0, 1/4]. We also show, via the degree Gini index, that uniform rooted binary trees are more regular than binary search trees grown from random permutations.

Keywords

chemical treecombinatorial probabilityGini indexrandom treetopological indexWiener index
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 34 , Issue 2 , April 2020 , pp. 157 - 171
Copyright
Copyright © Cambridge University Press 2019

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