## Abstract and Applied Analysis

### Some Properties on Estrada Index of Folded Hypercubes Networks

#### Abstract

Let $G$ be a simple graph with $n$ vertices and let ${\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{n}$ be the eigenvalues of its adjacency matrix; the Estrada index $EE(G)$ of the graph $G$ is defined as the sum of the terms ${e}^{\lambda i}$,  $i=1,2,\dots ,n$. The $n$-dimensional folded hypercube networks $F{Q}_{n}$ are an important and attractive variant of the $n$-dimensional hypercube networks ${Q}_{n}$, which are obtained from ${Q}_{n}$ by adding an edge between any pair of vertices complementary edges. In this paper, we establish the explicit formulae for calculating the Estrada index of the folded hypercubes networks $F{Q}_{n}$ by deducing the characteristic polynomial of the adjacency matrix in spectral graph theory. Moreover, some lower and upper bounds for the Estrada index of the folded hypercubes networks $F{Q}_{n}$ are proposed.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 167623, 6 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412278601

Digital Object Identifier
doi:10.1155/2014/167623

Mathematical Reviews number (MathSciNet)
MR3166571

Zentralblatt MATH identifier
07021848

#### Citation

Liu, Jia-Bao; Pan, Xiang-Feng; Cao, Jinde. Some Properties on Estrada Index of Folded Hypercubes Networks. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 167623, 6 pages. doi:10.1155/2014/167623. https://projecteuclid.org/euclid.aaa/1412278601

#### References

• J. M. Xu, Topological Strucure and Analysis of Interconnction Networks, Kluwer Academic, Dordrecht, The Netherlands, 2001.
• I. Gutman, “The energy of a graph,” Berichte der Mathematisch-Statistischen Sektion im Forschungszentrum Graz, vol. 103, no. 22, pp. 100–105, 1978.
• E. Estrada, “Characterization of 3D molecular structure,” Chemical Physics Letters, vol. 319, no. 5, pp. 713–718, 2000.
• E. Estrada and J. A. Rodríguez-Velázquez, “Subgraph centrality in complex networks,” Physical Review E, vol. 71, no. 5, Article ID 056103, 9 pages, 2005.
• E. Estrada and J. A. Rodríguez-Velázquez, “Spectral measures of bipartivity in complex networks,” Physical Review E, vol. 72, no. 4, Article ID 046105, 6 pages, 2005.
• K. C. Das, I. Gutman, and B. Zhou, “New upper bounds on Zagreb indices,” Journal of Mathematical Chemistry, vol. 46, no. 2, pp. 514–521, 2009.
• D. M. Cvetković, M. Doob, I. Gutman, and A. Torgašev, Recent Results in the Theory of Graph Spectra, vol. 36 of Annals of Discrete Mathematics, North-Holland, Amsterdam, The Netherlands, 1988.
• E. A. Amawy and S. Latifi, “Properties and performance of folded hypercubes,” IEEE Transactions on Parallel and Distributed Systems, vol. 2, no. 1, pp. 31–42, 1991.
• R. Indhumathi and S. A. Choudum, “Embedding certain height-balanced trees and complete ${p}^{m}$-ary trees into hypercubes,” Journal of Discrete Algorithms, vol. 22, pp. 53–65, 2013.
• J. Fink, “Perfect matchings extend to Hamilton cycles in hypercubes,” Journal of Combinatorial Theory B, vol. 97, no. 6, pp. 1074–1076, 2007.
• J. B. Liu, J. D. Cao, X. F. Pan, and A. Elaiw, “The Kirchhoff index of hypercubes and related complex networks,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 543189, 7 pages, 2013.
• M. Chen and B. X. Chen, “Spectra of folded hypercubes,” Journal of East China Normal University (Nature Science), vol. 2, pp. 39–46, 2011.
• X. He, H. Liu, and Q. Liu, “Cycle embedding in faulty folded hypercube,” International Journal of Applied Mathematics & Statistics, vol. 37, no. 7, pp. 97–109, 2013.
• S. Cao, H. Liu, and X. He, “On constraint fault-free cycles in folded hypercube,” International Journal of Applied Mathematics & Statistics, vol. 42, no. 12, pp. 38–44, 2013.
• Y. Zhang, H. Liu, and M. Liu, “Cycles embedding on folded hypercubes with vertex faults,” International Journal of Applied Mathematics & Statistics, vol. 41, no. 11, pp. 58–70, 2013.
• X. B. Chen, “Construction of optimal independent spanning trees on folded hypercubes,” Information Sciences, vol. 253, pp. 147–156, 2013.
• M. Chen, X. Guo, and S. Zhai, “Total chromatic number of folded hypercubes,” Ars Combinatoria, vol. 111, pp. 265–272, 2013.
• G. H. Wen, Z. S. Duan, W. W. Yu, and G. R. Chen, “Consensus in multi-agent systems with communication constraints,” International Journal of Robust and Nonlinear Control, vol. 22, no. 2, pp. 170–182, 2012.
• J. B. Liu, X. F. Pan, Y. Wang, and J. D. Cao, “The Kirchhoff index of folded hypercubes and some variant networks,” Mathematical Problems in Engineering, vol. 2014, Article ID 380874, 9 pages, 2014.
• E. Estrada, J. A. Rodríguez-Velázquez, and M. Randić, “Atomic branching in molecules,” International Journal of Quantum Chemistry, vol. 106, no. 4, pp. 823–832, 2006.
• B. Zhou and Z. Du, “Some lower bounds for Estrada index,” Iranian Journal of Mathematical Chemistry, vol. 1, no. 2, pp. 67–72, 2010.
• Y. Shang, “Lower bounds for the Estrada index of graphs,” Electronic Journal of Linear Algebra, vol. 23, pp. 664–668, 2012.
• J. P. Liu and B. L. Liu, “Bounds of the Estrada index of graphs,” Applied Mathematics, vol. 25, no. 3, pp. 325–330, 2010. \endinput