Extreme Wiener indices of trees with given number of vertices of maximum degree.

In this paper, we consider trees of order n with a given number of vertices of maximum degree and solve the minimal extremal problem for the Wiener index on that class. Characteristics of the extremal structures are presented, some directly from previously established methods. We also consider the maximal value of the Wiener index in the class of trees of order n with exactly k vertices of a given maximum degree, and characterize appropriate optimal trees. [ABSTRACT FROM AUTHOR]