On the sum of all distances in bipartite graphs

The transmission of a connected graph GG is the sum of all distances between all pairs of vertices in GG, it is also called the Wiener index of GG. In this paper, sharp bounds on the transmission are determined for several classes of connected bipartite graphs. For example, in the class of all connected nn-vertex bipartite graphs with a given matching number qq, the minimum transmission is realized only by the graph Kq,n−qKq,n−q; in the class of all connected nn-vertex bipartite graphs of diameter dd, the extremal graphs with the minimal transmission are characterized. Moreover, all the extremal graphs having the minimal transmission in the class of all connected nn-vertex bipartite graphs with a given vertex connectivity (resp. edge-connectivity) are also identified.