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An added edge to a graph is called an inset edge. Predicting k inset edgeswhich minimize the average distance of a graph is known to be NP-Hard. However,when k = 1 the complexity of the problem is polynomial. In this paper, sometools for a precise analysis of the problem for the trees are established.Using the tools, we can avoid using the distance matrix. This leads to moreefficient algorithms and a better analysis of the problem. Several applicationsof the tools as well as a tight bound for the change of average distance whenan inset edge is added to a tree are presented.

Some Preliminary Result About the Inset Edge and Average Distance of Trees