Investigating Several Fundamental Properties of Random Lobster Trees and Random Spider Trees

Probabilistic Methods in the Study of Topological Indices on Random Spider Trees

“…In a very recent article (Ren, Zhang, and Dey 2021), the authors inspired by the seminal paper (Barabási and Albert 1999) introduced a class of RSTs that evolves in a preferential attachment manner as follows. At time 1, a RST starts with a seed graph containing a centroid of degree 3 and three leaves.…”

“…This index is a topological measure on a graph capturing the proximity to regular graphs. In (Ren, Zhang, and Dey 2021) the authors considered the degreebased Gini index introduced by (Domicolo and Mahmoud 2020), with slight modifications. In this section, we study the degree-based Gini index defined in (Ren, Zhang, and Dey 2021).…”

“…For a better treatment we refer interested readers to (Martínez-Martínez, Méndez-Bermúdez, Rodríguez, and Sigarreta 2020;Aguilar-Sánchez, Méndez-Bermúdez, Rodríguez, and Sigarreta 2021;Zhang and Wang 2021;Li, Shi, and Gao 2020;Li, Shi, and Gao 2021;Kazemi 2021;Pegu, Deka, Gogoi, and Bharali 2021). In particular, motivated by the substantial increase of interests in random tree models Ren, Zhang and Dey investigated two classes of random lobster trees which evolve according to different rules and a class of random spider trees which grow in a preferential attachment manner (Ren, Zhang, and Dey 2021). In this important work, the authors obtained very useful results, as a matter of fact, for the class of random spider trees they characterized the structure of the model by determining the exact and asymptotic distributions of the number of leaves, and by computing two kinds of topological indices: Zagreb index and Gini index.…”

“…In this important work, the authors obtained very useful results, as a matter of fact, for the class of random spider trees they characterized the structure of the model by determining the exact and asymptotic distributions of the number of leaves, and by computing two kinds of topological indices: Zagreb index and Gini index. Inspired by (Ren, Zhang, and Dey 2021) and considering the arguments put forward in the previous paragraphs, in this manuscript, we consider a class of spider trees that are incorporated with randomness, called random spiders trees (RSTs), and we investigate several useful topological indices of this random class, including degree-based Gini index, degree-based Hoover index, generalized Zagreb index and other indices associated with these. Specifically, a central limit theorem is developed for the asymptotic distribution of the number of leaves of a RST.…”

Single Imputation Methods and Confidence Intervals for the Gini Index

“…An excellent review of the Gini index can be found in [25]. It is important to note that the Gini index and related mea-sures have also been adopted in other contexts, such as for the construction of topological indices for trees and graphs (see [26,27]), for the analysis of reliability systems ( [28]) and for constructing decision-making methods ( [29]). A key advantage of the Gini index is its ease of interpretation, as it takes values between 0 and 1, where 0 indicates perfect equality and 1 the opposite.…”