Bounds for the Kirchhoff Index of Bipartite Graphs
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...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2012-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2012/195242 |
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| Summary: | 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. |
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| ISSN: | 1110-757X 1687-0042 |