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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 543189, 7 pages
http://dx.doi.org/10.1155/2013/543189
Research Article

The Kirchhoff Index of Hypercubes and Related Complex Networks

1Department of Mathematics, Southeast University, Nanjing 210096, China
2School of Mathematical Sciences, Anhui University, Hefei 230601, China
3Anhui Xinhua University, Hefei 230088, China
4Department of Mathematics, Faculty of Sciences, King Abdulaziz University, Jeddah, Saudi Arabia

Received 14 October 2013; Accepted 31 October 2013

Academic Editor: Guanghui Wen

Copyright © 2013 Jiabao Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The resistance distance between any two vertices of is defined as the network effective resistance between them if each edge of is replaced by a unit resistor. The Kirchhoff index Kf() is the sum of resistance distances between all the pairs of vertices in . We firstly provided an exact formula for the Kirchhoff index of the hypercubes networks by utilizing spectral graph theory. Moreover, we obtained the relationship of Kirchhoff index between hypercubes networks and its three variant networks , , by deducing the characteristic polynomial of the Laplacian matrix related networks. Finally, the special formulae for the Kirchhoff indexes of , , and were proposed, respectively.