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29 March 2024
 
  » arxiv » 1005.1135

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The asymptotic number of occurrences of a subtree in trees with bounded maximum degree and an application to the Estrada index
Xueliang LI ; Yiyang Li ;
Date 7 May 2010
AbstractLet $mathcal {T}^{Delta}_n$ 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.
Source arXiv, 1005.1135
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