# Lower Bounds for Gaussian Estrada Index of Graphs

## Abstract

**:**

## 1. Introduction

## 2. Results and Discussion

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Remark**

**1.**

**Theorem**

**4.**

**Proof.**

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

## 3. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Cvetković, D.M.; Doob, M.; Gutman, I.; Torgašev, A. Recent Results in the Theory of Graph Spectra; North-Holland: Amsterdam, The Netherlands, 1988. [Google Scholar]
- Estrada, E. Characterization of 3D molecular structure. Chem. Phys. Lett.
**2000**, 319, 713–718. [Google Scholar] [CrossRef] - Estrada, E. Characterization of the folding degree of proteins. Bioinformatics
**2002**, 18, 697–704. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Estrada, E. Characterization of the amino acid contribution to the folding degree of proteins. Proteins
**2004**, 54, 727–737. [Google Scholar] [CrossRef] [PubMed] - Estrada, E.; Rodríguez-Velázquez, J.A. Subgraph centrality in complex networks. Phys. Rev. E
**2005**, 71, 056103. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Estrada, E.; Rodríguez-Velázquez, J.A. Spectral measures of bipartivity in complex networks. Phys. Rev. E
**2005**, 72, 046105. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Estrada, E.; Rodríguez-Velázquez, J.A.; Randić, M. Atomic branching in molecules. Int. J. Quantum Chem.
**2006**, 106, 823–832. [Google Scholar] [CrossRef] - Shang, Y. Local natural connectivity in complex networks. Chin. Phys. Lett.
**2011**, 28, 068903. [Google Scholar] [CrossRef] - Shang, Y. Biased edge failure in scale-free networks based on natural connectivity. Indian J. Phys.
**2012**, 86, 485–488. [Google Scholar] [CrossRef] - Wu, J.; Barahona, M.; Tan, Y.J.; Deng, H.Z. Robustness of regular ring lattices based on natural connectivity. Int. J. Syst. Sci.
**2011**, 42, 1085–1092. [Google Scholar] [CrossRef] [Green Version] - Gutman, I. Lower bounds for Estrada index. Publ. Inst. Math. Beograd (N.S.)
**2008**, 83, 1–7. [Google Scholar] [CrossRef] - Gutman, I.; Deng, H.; Radenković, S. The Estrada index: an updated survey. In Selected Topics on Applications of Graph Spectra; Cvetković, D., Gutman, I., Eds.; Mathematical Institute: Belgrade, Serbia, 2011; pp. 155–174. [Google Scholar]
- Gutman, I.; Radenković, S. A lower bound for the Estrada index of bipartite molecular graphs. Kragujev. J. Sci.
**2007**, 29, 67–72. [Google Scholar] - De la Peña, J.A.; Gutman, I.; Rada, J. Estimating the Estrada index. Lin. Algebra Appl.
**2007**, 427, 70–76. [Google Scholar] [CrossRef] - Shang, Y. Lower bounds for the Estrada index of graphs. Electron. J. Linear Algebra
**2012**, 23, 664–668. [Google Scholar] [CrossRef] - Shang, Y. Estrada index of general weighted graphs. Bull. Aust. Math. Soc.
**2013**, 88, 106–112. [Google Scholar] [CrossRef] - Zhou, B. On Estrada index. MATCH Commun. Math. Comput. Chem.
**2008**, 60, 485–492. [Google Scholar] - Lenes, E.; Mallea-Zepeda, E.; Robbiano, M.; Rodríguez, J. On the diameter and incidence energy of iterated total graphs. Symmetry
**2018**, 10, 252. [Google Scholar] [CrossRef] - Borovićanin, B.; Gutman, I. Nullity of graphs: an updated survey. In Selected Topics on Applications of Graph Spectra; Cvetković, D., Gutman, I., Eds.; Mathematical Institute: Belgrade, Serbia, 2011; pp. 137–154. [Google Scholar]
- Kutzelnigg, W. What I like about Hückel theory. J. Comput. Chem.
**2007**, 28, 25–34. [Google Scholar] [CrossRef] [PubMed] - Estrada, E.; Alhomaidhi, A.A.; Al-Thukair, F. Exploring the “Middel Earth” of network spectra via a Gaussian matrix function. Chaos
**2017**, 27, 023109. [Google Scholar] [CrossRef] [PubMed] - Wang, L.W.; Zunger, A. Solving Schrödinger’s equation around a desired energy: application to silicon quantum dots. J. Chem. Phys.
**1994**, 100, 2394. [Google Scholar] [CrossRef] - Gutman, I.; Trinajstić, N. Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons. Chem. Phys. Lett.
**1972**, 17, 535–538. [Google Scholar] [CrossRef] - Gutman, I.; Das, K.C. The first Zagreb index 30 years after. MATCH Commun. Math. Comput. Chem.
**2004**, 50, 83–92. [Google Scholar] - Shao, Z.; Siddiqui, M.K.; Muhammad, M.H. Computing Zagreb indices and Zagreb polynomials for symmetrical nanotubes. Symmetry
**2018**, 10, 244. [Google Scholar] [CrossRef] - Zhou, B. On spectral radius of nonnegative matrices. Australas. J. Combin.
**2000**, 22, 301–306. [Google Scholar] - Shang, Y. Estrada index of random bipartite graphs. Symmetry
**2015**, 7, 2195–2205. [Google Scholar] [CrossRef] - Shang, Y. The Estrada index of evolving graphs. Appl. Math. Comput.
**2015**, 250, 415–423. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Shang, Y.
Lower Bounds for Gaussian Estrada Index of Graphs. *Symmetry* **2018**, *10*, 325.
https://doi.org/10.3390/sym10080325

**AMA Style**

Shang Y.
Lower Bounds for Gaussian Estrada Index of Graphs. *Symmetry*. 2018; 10(8):325.
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**Chicago/Turabian Style**

Shang, Yilun.
2018. "Lower Bounds for Gaussian Estrada Index of Graphs" *Symmetry* 10, no. 8: 325.
https://doi.org/10.3390/sym10080325