The asymptotic values of the general Zagreb and Randi\'c indices of trees with bounded maximum degree

TL;DR

This paper analyzes the asymptotic behavior of Zagreb and Randić indices in bounded-degree trees, showing that degree distributions and edge types follow normal distributions with predictable means and variances.

Contribution

It provides the first asymptotic normality results for degree and edge distributions in bounded-degree trees, enabling estimates of Zagreb and Randić indices for large random trees.

Findings

Degree counts are asymptotically normal with linear mean and variance.

Edge type counts follow a normal distribution with linear mean and variance.

Explicit estimates for Zagreb and Randić indices in large random trees.

Abstract

Let $\mathcal {T}^{\Delta}_n$ denote the set of trees of order $n$, in which the degree of each vertex is bounded by some integer $\Delta$. Suppose that every tree in $\mathcal {T}^{\Delta}_n$ is equally likely. We show that the number of vertices of degree $j$ in $\mathcal {T}^{\Delta}_n$ is asymptotically normal with mean $(\mu_j+o(1))n$ and variance $(\sigma_j+o(1))n$, where $\mu_j$, $\sigma_j$ are some constants. As a consequence, we give estimate to the value of the general Zagreb index for almost all trees in $\mathcal {T}^{\Delta}_n$. Moreover, we obtain that the number of edges of type $(i,j)$ in $\mathcal {T}^{\Delta}_n$ also has mean $(\mu_{ij}+o(1))n$ and variance $(\sigma_{ij}+o(1))n$, where an edge of type $(i,j)$ means that the edge has one end of degree $i$ and the other of degree $j$, and $\mu_{ij}$, $\sigma_{ij}$ are some constants. Then, we give estimate to the value of the general Randi\'{c} index for almost all trees in $\mathcal {T}^{\Delta}_n$.