# Estrada and L-Estrada Indices of Edge-Independent Random Graphs

## Abstract

**:**

## 1. Introduction

## 2. Estrada Index of Random Graphs

**Lemma 1.**[24] Consider a random graph ${G}_{n}\left({p}_{ij}\right)$. If $\Delta \gg {(lnn)}^{4}$, then

**Theorem 1.**If $\Delta \gg {(lnn)}^{4}$, then

**Proof**. It follows from Lemma 1 that

**Theorem 2.**Consider a random graph ${G}_{n}\left(p\right)$ with $p\in (0,1)$. We have

**Proof**. Recalling the definition of ${P}_{n}$, we have now ${P}_{n}=p({J}_{n}-{I}_{n})$, where ${J}_{n}\in {\mathbb{R}}^{n\times n}$ is the matrix whose all entries equal 1. By the Chernoff bound, it holds that $\Delta =(1+o(1\left)\right)np$ a.s. Since the eigenvalues of ${P}_{n}$ are ${\lambda}_{1}\left({P}_{n}\right)=p(n-1)$, ${\lambda}_{2}\left({P}_{n}\right)=\cdots ={\lambda}_{n}\left({P}_{n}\right)=-p$, Theorem 1 immediately implies that

**Corollary 1.**Consider a random graph ${G}_{n}^{o}\left(p\right)$ with $p\in (0,1)$. We have

**Figure 1.**Logarithmic Estrada index $ln\left(EE\left({G}_{n}\left(p\right)\right)\right)$ versus connection probability p for two different graph sizes of $n=$ 4000 and 6000. Theoretical bounds (solid and dashed curves) are from Theorem 2. Simulated results (circles and crosses) are obtained by means of an ensemble averaging of 100 randomly generated graphs yielding a statistically ample enough sampling.

## 3. 𝓛-Estrada Index of Random Graphs

**Lemma 2.**[24] Consider a random graph ${G}_{n}\left({p}_{ij}\right)$. If $\mathrm{rank}\left({P}_{n}\right)=r$ and $\delta \gg max\{r,{(lnn)}^{4}\}$, then

**Theorem 3.**If $\mathrm{rank}\left({P}_{n}\right)=r$ and $\delta \gg max\{r,{(lnn)}^{4}\}$, then

**Proof**. Thanks to Lemma 2, we have

**Theorem 4.**Consider a random graph ${G}_{n}^{o}\left(p\right)$ with $p\in (0,1)$. We have

**Proof of Theorem 4**. First note that $\delta =(1+o(1\left)\right)np$ a.s. by the Chernoff bound. Since ${P}_{n}=p{J}_{n}$, we have $\mathrm{rank}\left({P}_{n}\right)=1\ll \delta $. Noting that ${L}_{n}={I}_{n}-{T}_{n}^{-1/2}{P}_{n}{T}_{n}^{-1/2}={I}_{n}-\frac{1}{n}{J}_{n}$, the eigenvalues of ${L}_{n}$ are ${\lambda}_{1}\left({L}_{n}\right)=\cdots ={\lambda}_{n-1}\left({L}_{n}\right)=1$ and ${\lambda}_{n}\left({L}_{n}\right)=0$. From Theorem 3, we readily conclude that

**Figure 2.**Logarithmic $\U0001d4db$-Estrada index $ln\left(\U0001d4dbEE\left({G}_{n}^{o}\left(p\right)\right)\right)$ versus connection probability p for two different graph sizes of $n=$ 4000 and 6000. Theoretical bounds (solid and dashed curves) are from Theorem 4. Simulated results (circles and crosses) are obtained by means of an ensemble averaging of 100 randomly generated graphs yielding a statistically ample enough sampling.

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

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Shang, Y.
Estrada and *L*-Estrada Indices of Edge-Independent Random Graphs. *Symmetry* **2015**, *7*, 1455-1462.
https://doi.org/10.3390/sym7031455

**AMA Style**

Shang Y.
Estrada and *L*-Estrada Indices of Edge-Independent Random Graphs. *Symmetry*. 2015; 7(3):1455-1462.
https://doi.org/10.3390/sym7031455

**Chicago/Turabian Style**

Shang, Yilun.
2015. "Estrada and *L*-Estrada Indices of Edge-Independent Random Graphs" *Symmetry* 7, no. 3: 1455-1462.
https://doi.org/10.3390/sym7031455