An inverse result for Wang's theorem on extremal trees

It was recently noted by Damnjanovic et al. [MATCH Commun. Math. Comput. Chem. 90 (2023), 197-202] that the problem of finding a tree which minimises or maximises the Sombor index among all the trees with a given degree sequence fits within the framework of results by Hua Wang from [Cent. Eur. J. Math. 12 (2014), 1656-1663]. Here, we extend these results by providing an inverse for the aforementioned theorem by Wang. In other words, for any fixed symmetric function f satisfying a monotonicity condition thatf (x, a) + f(y, b) > f(y, a) + f (x, b) for any x > y and a > b,we characterise precisely the set of all the trees minimising or maximising the sum f (deg x, deg y) over all the adjacent pairs of vertices x and y, among the trees with a given degree sequence.