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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 195242, 9 pages
http://dx.doi.org/10.1155/2012/195242
Research Article

Bounds for the Kirchhoff Index of Bipartite Graphs

School of Mathematics and Information Science, Yantai University, Yantai 264005, China

Received 1 January 2012; Accepted 10 February 2012

Academic Editor: Mehmet Sezer

Copyright © 2012 Yujun Yang. 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

A -bipartite graph is a bipartite graph such that one bipartition has m vertices and the other bipartition has n vertices. The tree dumbbell consists of the path together with a independent vertices adjacent to one pendent vertex of and b independent vertices adjacent to the other pendent vertex of . In this paper, firstly, we show that, among -bipartite graphs , the complete bipartite graph has minimal Kirchhoff index and the tree dumbbell has maximal Kirchhoff index. Then, we show that, among all bipartite graphs of order , the complete bipartite graph has minimal Kirchhoff index and the path has maximal Kirchhoff index, respectively. Finally, bonds for the Kirchhoff index of -bipartite graphs and bipartite graphs of order are obtained by computing the Kirchhoff index of these extremal graphs.