Abstract
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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