Year: 2025
Type: article
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 k th-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.
Source: Advances in Applied Probability
DOI: 10.1017/apr.2025.19
Authors: Qunqiang Feng, Hongpeng Ren, Yaru Tian
Institution: University of Science and Technology of China
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4410755484