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Limit laws for the generalized Zagreb indices of random graphs

Part of: Graph theory Limit theorems Combinatorial probability

Published online by Cambridge University Press:  26 May 2025

Qunqiang Feng Open the ORCID record for Qunqiang Feng [Opens in a new window] ,
Hongpeng Ren  and
Yaru Tian
Qunqiang Feng*
Affiliation:
University of Science and Technology of China
Hongpeng Ren*
Affiliation:
University of Science and Technology of China
Yaru Tian*
Affiliation:
University of Science and Technology of China
*
*Postal address: Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei 230026, China.
*Postal address: Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei 230026, China.
*Postal address: Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei 230026, China.
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Abstract

In this paper, we study the asymptotic behavior of the generalized Zagreb indices of the classical Erdős–Rényi (ER) random graph G(n, p), as $n\to\infty$. For any integer $k\ge1$, we first give an expression for the kth-order generalized Zagreb index in terms of the number of star graphs of various sizes in any simple graph. The explicit formulas for the first two moments of the generalized Zagreb indices of an ER random graph are then obtained from this expression. Based on the asymptotic normality of the numbers of star graphs of various sizes, several joint limit laws are established for a finite number of generalized Zagreb indices with a phase transition for p in different regimes. Finally, we provide a necessary and sufficient condition for any single generalized Zagreb index of G(n, p) to be asymptotic normal.

Keywords

Erdős–Rényi Random graphtopological indexStirling numberstar graphdependency graph

MSC classification

Primary: 05C80: Random graphs 60C05: Combinatorial probability
Secondary: 60F05: Central limit and other weak theorems
Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 20
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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