Bounds for the kirchhoff index of bipartite graphs

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 p n - a - b together with a independent vertices adjacent to one pendent vertex of p n - a - b and b independent vertices adjacent to the other pendent vertex of p n - a - b. in this paper,firstly,we show that,among (m,n) -bipartite graphs (m n),the complete bipartite graph k m,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 p l 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. copyright © 2012 yujun yang.