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On the sum of all distances in bipartite graphs. (English) Zbl 1288.05072
Summary: The transmission of a connected graph $$G$$ is the sum of all distances between all pairs of vertices in $$G$$, it is also called the Wiener index of $$G$$. 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 $$n$$-vertex bipartite graphs with a given matching number $$q$$, the minimum transmission is realized only by the graph $$K_{q,n-q}$$; in the class of all connected $$n$$-vertex bipartite graphs of diameter $$d$$, the extremal graphs with the minimal transmission are characterized. Moreover, all the extremal graphs having the minimal transmission in the class of all connected $$n$$-vertex bipartite graphs with a given vertex connectivity (resp. edge-connectivity) are also identified.

##### MSC:
 05C12 Distance in graphs 05C40 Connectivity
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