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Bounds for the Kirchhoff Index of Bipartite Graphs

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  • Yujun Yang

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.

Suggested Citation

  • Yujun Yang, 2012. "Bounds for the Kirchhoff Index of Bipartite Graphs," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-9, April.
    • Handle: RePEc:hin:jnljam:195242
    DOI: 10.1155/2012/195242
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