# On Generalized Distance Gaussian Estrada Index of Graphs

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## Abstract

**:**

## 1. Introduction

## 2. Motivation

## 3. Bounds for Generalized Distance Gaussian Estrada Index

**Lemma**

**1.**

**Lemma**

**2.**

**Lemma**

**3.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Corollary**

**1.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Theorem**

**4.**

**Proof.**

- (i)
- G has exactly one distinct ${D}_{\alpha}$-eigenvalue. Then $G={K}_{1}.$
- (ii)
- G has exactly two distinct ${D}_{\alpha}$-eigenvalues. Then, by Lemma 3, $G={K}_{n}.$
- (iii)
- G has exactly three distinct ${D}_{\alpha}$-eigenvalues. Then ${\partial}_{1}=\sqrt{\frac{{\sum}_{i=1}^{n}{(\alpha T{r}_{i}^{2}+(1-\alpha ){T}_{i})}^{2}}{{\sum}_{i=1}^{n}T{r}_{i}^{2}}}$ and $\left|{\partial}_{i}\right|=\sqrt{\frac{P-\frac{{\sum}_{i=1}^{n}{(\alpha T{r}_{i}^{2}+(1-\alpha ){T}_{i})}^{2}}{{\sum}_{i=1}^{n}T{r}_{i}^{2}}}{n-1}},\phantom{\rule{1.em}{0ex}}i=2,\dots ,n$. Moreover, for $\frac{1}{2}\le \alpha \le 1,$ G is k-transmission regular graph. Then it is clear that G is a graph with exactly three distinct ${D}_{\alpha}$-eigenvalues$$\left(2k,\sqrt{\frac{M+{k}^{2}(n{\alpha}^{2}-4)}{n-1}},-\sqrt{\frac{M+{k}^{2}(n{\alpha}^{2}-4)}{n-1}}\right),$$

**Theorem**

**5.**

**Proof.**

**Remark**

**2.**

**Lemma**

**6.**

**Theorem**

**6.**

**Proof.**

**Remark**

**3.**

**Corollary**

**3.**

## 4. Examples for Some Fundamental Special Graphs

**Lemma**

**7.**

**Lemma**

**8.**

**Corollary**

**4.**

**Corollary**

**5.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Definition**

**1.**

- (a)
- ${u}_{1}$ is adjacent to ${v}_{1}$ in G or
- (b)
- ${u}_{1}={v}_{1}$ and ${u}_{2}$ is adjacent to ${v}_{2}$ in G.

**Example**

**4.**

**Example**

**5.**

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Cvetković, D.M.; Doob, M.; Sachs, H. Spectra of Graphs—Theory and Application; Academic Press: New York, NY, USA, 1980. [Google Scholar]
- Aouchiche, M.; Hansen, P. Two Laplacians for the distance matrix of a graph. Linear Algebra Appl.
**2013**, 439, 21–33. [Google Scholar] [CrossRef] - Alhevaz, A.; Baghipur, M.; Hashemi, E.; Ramane, H.S. On the distance signless Laplacian spectrum of graphs. Bull. Malay. Math. Sci. Soc.
**2019**, 42, 2603–2621. [Google Scholar] [CrossRef] - Alhevaz, A.; Baghipur, M.; Paul, S. On the distance signlees Laplacian spectral radius and the distance signless Laplacian energy of graphs. Discrete Math. Algorithms Appl.
**2018**, 10, 19. [Google Scholar] [CrossRef] - Xing, R.; Zhou, B.; Li, J. On the distance signless Laplacian spectral radius of graphs. Linear Multilinear Algebra
**2014**, 62, 1377–1387. [Google Scholar] [CrossRef] - Nikiforov, V. Merging the A- and Q-spectral theories. Appl. Anal. Discrete Math.
**2017**, 11, 81–107. [Google Scholar] [CrossRef] - Cui, S.Y.; He, J.X.; Tian, G.X. The generalized distance matrix. Linear Algebra Appl.
**2019**, 563, 1–23. [Google Scholar] [CrossRef] - Cvetković, D.M. Applications of graph spectra: An introduction to the literature. Appl. Graph Spectra
**2009**, 21, 7–31. [Google Scholar] - Estrada, E. Characterization of the amino acid contribution to the folding degree of proteins. Proteins
**2004**, 54, 727–737. [Google Scholar] [CrossRef] - Gutman, I.; Estrada, E.; Rodriguez-Velázquez, J.A. On a graph-spectrum-based structure descriptor. Croat. Chem. Acta
**2007**, 80, 151–154. [Google Scholar] - De la Peña, J.A.; Gutman, I.; Rada, J. Estimating the Estrada index. Linear Algebra Appl.
**2007**, 427, 70–76. [Google Scholar] [CrossRef] [Green Version] - Gutman, I.; Furtula, B.; Chen, X.; Qian, J. Resolvent Estrada index—Computational and mathematical studies. Match Commun. Math. Comput. Chem.
**2015**, 74, 431–440. [Google Scholar] - Gutman, I.; Deng, H.; Radenković, S. The Estrada index: An updated survey. Sel. Top. Appl. Graph Spectra
**2011**, 22, 155–174. [Google Scholar] - Estrada, E. Characterization of 3-D molecular structure. Chem. Phys. Lett.
**2000**, 319, 713–718. [Google Scholar] [CrossRef] - Estrada, E.; Rodriguez-Velázguez, 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] - Estrada, E. The Structure of Complex Networks-Theory and Applications; Oxford Univ. Press: New York, NY, USA, 2012. [Google Scholar]
- Ayyaswamy, S.K.; Balachandran, S.; Venkatakrishnan, Y.B.; Gutman, I. Signless Laplacian Estrada index. Match Commun. Math. Comput. Chem.
**2011**, 66, 785–794. [Google Scholar] - Güngör, A.D.; Bozkurt, Ş.B. On the distance Estrada index of graphs. Hacet. J. Math. Stat.
**2009**, 38, 277–283. [Google Scholar] - Ilić, A.; Zhou, B. Laplacian Estrada index of trees. Match Commun. Math. Comput. Chem.
**2010**, 63, 769–776. [Google Scholar] - Shang, Y. Distance Estrada index of random graphs. Linear Multilinear Algebra
**2015**, 63, 466–471. [Google Scholar] [CrossRef] - Shang, Y. Bounds of distance Estrada index of graphs. Ars Combin.
**2016**, 128, 287–294. [Google Scholar] - Alhomaidhi, A.A.; Al-Thukair, F.; Estrada, E. Gaussianization of the spectra of graphs and networks. Theory and applications. J. Math. Anal. Appl.
**2019**, 470, 876–897. [Google Scholar] [CrossRef] - 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 silico quantum dots. J. Chem. Phys.
**1994**, 100, 2394. [Google Scholar] [CrossRef] - Shang, Y. Lower bounds for Gaussian Estrada index of graphs. Symmetry
**2018**, 10, 325. [Google Scholar] [CrossRef] - Aouchiche, M.; Hansen, P. Distance spectra of graphs: A survey. Linear Algebra Appl.
**2014**, 458, 301–386. [Google Scholar] [CrossRef] - Indulal, G. Sharp bounds on the distance spectral radius and the distance energy of graphs. Linear Algebra Appl.
**2009**, 430, 106–113. [Google Scholar] [CrossRef] [Green Version] - Minć, H. Nonnegative Matrices; John Wiley and Sons Inc.: New York, NY, USA, 1988. [Google Scholar]
- Diaz, R.C.; Rojo, O. Sharp upper bounds on the distance energies of a graph. Linear Algebra Appl.
**2018**, 545, 55–75. [Google Scholar] [CrossRef] - Zhou, B.; Gutman, I.; Aleksić, T. A note on Laplacian energy of graphs. Match Commun. Math. Comput. Chem.
**2008**, 60, 441–446. [Google Scholar] - Ganie, H.A.; Pirzada, S.; Alhevaz, A.; Baghipur, M. Generalized distance spectral spread of a graph.
**2019**. submitted. [Google Scholar] - Indulal, G.; Gutman, I.; Vijayakumar, A. On distance energy of graphs. Match Commun. Math. Comput. Chem.
**2008**, 60, 461–472. [Google Scholar] - Indulal, G. The distance spectrum of graph compositions. Ars. Math. Contemp.
**2009**, 2, 93–100. [Google Scholar]

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Alhevaz, A.; Baghipur, M.; Shang, Y.
On Generalized Distance Gaussian Estrada Index of Graphs. *Symmetry* **2019**, *11*, 1276.
https://doi.org/10.3390/sym11101276

**AMA Style**

Alhevaz A, Baghipur M, Shang Y.
On Generalized Distance Gaussian Estrada Index of Graphs. *Symmetry*. 2019; 11(10):1276.
https://doi.org/10.3390/sym11101276

**Chicago/Turabian Style**

Alhevaz, Abdollah, Maryam Baghipur, and Yilun Shang.
2019. "On Generalized Distance Gaussian Estrada Index of Graphs" *Symmetry* 11, no. 10: 1276.
https://doi.org/10.3390/sym11101276