The Asymptotic Behavior of the Estrada Index for Trees

Xueliang Li ; Yiyang Li

Bulletin of the Malaysian Mathematical Society, Tome 36 (2013) no. 1

Résumé

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 small tree $H$, we first show that the number of occurrences of $H$ in trees of $\mathcal {T}^{\Delta}_n$ has mean $(\mu_H+o(1))n$ and variance $(\sigma_H+o(1))n$, where $\mu_H$, $\sigma_H$ are some constants. Then we apply this result to estimate the value of the Estrada index $EE$ for almost all trees in $\mathcal {T}^{\Delta}_n$, and give a theoretical explanation to the approximate linear correlation between $EE$ and the first Zagreb index obtained by quantitative analysis.

Détail

Classification : 05C05, 05C12, 05C30, 05D40, 05A15, 05A16, 92E10