An inverse result for Wang's theorem on extremal trees

Abstract

It was recently noted by Damnjanović 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 that
\[
f(x, a) + f(y, b) > f(y, a) + f(x, b) \quad \mbox{for any $x > y$ and $a > b$} ,
\]
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.