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Mathematical Problems in Engineering
Volume 2014 (2014), Article ID 286876, 8 pages
http://dx.doi.org/10.1155/2014/286876
Research Article

The Kirchhoff Index of Toroidal Meshes and Variant Networks

1School of Mathematical Sciences, Anhui University, Hefei 230601, China
2Department of Mathematics, Southeast University, Nanjing 210096, China
3Department of Public Courses, Anhui Xinhua University, Hefei 230088, China
4Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
5College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China

Received 14 March 2014; Accepted 20 May 2014; Published 3 June 2014

Academic Editor: He Huang

Copyright © 2014 Jia-Bao 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 is a novel distance function on electrical network theory proposed by Klein and Randić. The Kirchhoff index Kf( ) is the sum of resistance distances between all pairs of vertices in . In this paper, we established the relationships between the toroidal meshes network and its variant networks in terms of the Kirchhoff index via spectral graph theory. Moreover, the explicit formulae for the Kirchhoff indexes of , , , and were proposed, respectively. Finally, the asymptotic behavior of Kirchhoff indexes in those networks is obtained by utilizing the applications of analysis approach.