Bounds for the Kirchhoff Index of Bipartite Graphs
نویسنده
چکیده
A m,n -bipartite graph is a bipartite graph such that one bipartition has m vertices and the other bipartition has n vertices. The tree dumbbell D n, a, b consists of the path Pn−a−b together with a independent vertices adjacent to one pendent vertex of Pn−a−b and b independent vertices adjacent to the other pendent vertex of Pn−a−b. In this paper, firstly, we show that, among m,n bipartite graphs m ≤ n , the complete bipartite graph Km,n has minimal Kirchhoff index and the tree dumbbell D m n, n − m 1 /2 , n − m 1 /2 has maximal Kirchhoff index. Then, we show that, among all bipartite graphs of order l, the complete bipartite graph K l/2 ,l− l/2 has minimal Kirchhoff index and the path Pl has maximal Kirchhoff index, respectively. Finally, bonds for the Kirchhoff index of m,n -bipartite graphs and bipartite graphs of order l are obtained by computing the Kirchhoff index of these extremal graphs.
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ورودعنوان ژورنال:
- J. Applied Mathematics
دوره 2012 شماره
صفحات -
تاریخ انتشار 2012