Bounds for the Kirchhoff Index of Bipartite Graphs

Abstract

A $\left(m,n\right)$-bipartite graph is a bipartite graph such that one bipartition has m vertices and the other bipartition has n vertices. The tree dumbbell $D\left(n,a,b\right)$ 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 $\left(m,n\right)$-bipartite graphs $\left(m\le n\right)$, the complete bipartite graph $K_{m,n}$ has minimal Kirchhoff index and the tree dumbbell $D\left(m+n,n-\mathrm{(m}+\mathrm{1)}/2,n-\mathrm{(m}+\mathrm{1)}/2\right)$ has maximal Kirchhoff index. Then, we show that, among all bipartite graphs of order $l$, the complete bipartite graph $K_{\{\backslash lfloor\}l/2\{\backslash rfloor\},l-\{\backslash lfloor\}l/2\{\backslash rfloor\}}$ has minimal Kirchhoff index and the path $P_l$ has maximal Kirchhoff index, respectively. Finally, bonds for the Kirchhoff index of $\left(m,n\right)$-bipartite graphs and bipartite graphs of order $l$ are obtained by computing the Kirchhoff index of these extremal graphs.