The asymptotic number of occurrences of a subtree in trees with bounded maximum degree and an application to the Estrada index

TL;DR

This paper analyzes the asymptotic frequency of subtrees in bounded-degree trees and applies findings to estimate the Estrada index, revealing a theoretical basis for its correlation with the Zagreb index.

Contribution

It provides the first asymptotic formulas for subtree occurrences in bounded-degree trees and links these results to Estrada index estimation and correlation analysis.

Findings

Expected subtree occurrences grow linearly with tree size

Estrada index can be estimated for almost all bounded-degree trees

Linear correlation between Estrada index and Zagreb index explained theoretically

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. For any given subtree $H$, we show that the number of occurrences of $H$ in trees of $\mathcal {T}^{\Delta}_n$ is with mean $(\mu_H+o(1))n$ and variance $(\sigma_H+o(1))n$, where $\mu_H$, $\sigma_H$ are some constants. As an application, we estimate the value of the Estrada index $EE$ for almost all trees in $\mathcal {T}^{\Delta}_n$, and give an explanation in theory to the approximate linear correlation between $EE$ and the first Zagreb index obtained by quantitative analysis.