An inverse result for Wang's theorem on extremal trees

TL;DR

This paper characterizes the extremal trees that maximize or minimize the sum of a symmetric, monotonic function over adjacent vertex degrees, extending Wang's theorem on extremal trees.

Contribution

It provides a precise characterization of trees that are extremal for sums involving a symmetric, monotonic function of degrees, solving the inverse problem of Wang's theorem.

Findings

Greedy trees maximize the sum for the given function.

Alternating greedy trees minimize the sum.

The paper offers a complete classification of extremal trees for these problems.

Abstract

Among all trees on $n$ vertices with a given degree sequence, how do we maximise or minimise the sum over all adjacent pairs of vertices $x$ and $y$ of $f(\mathrm{deg} x, \mathrm{deg} y)$? Here $f$ is a fixed symmetric function satisfying a 'monotonicity' condition that \[ f(x, a) + f(y, b) > f(y, a) + f(x, b) \quad \mbox{for any $x > y$ and $a > b$} . \] These functions arise naturally in several areas of graph theory, particularly chemical graph theory. Wang showed that the so-called 'greedy' tree maximises this quantity, while an 'alternating greedy' tree minimises it. Our aim in this paper is to solve the inverse problem: we characterise precisely which trees are extremal for these two problems.