An algorithmic approach to extending a theorem on extremal trees by Wang

Abstract

Let $\mathcal{T}_D$ be the set containing all the trees corresponding to a given degree sequence $D$ and let $R_\mathcal{F}$ be the graph invariant defined via $$R_\mathcal{F} = \sum_{u \sim v} \mathcal{F}(\mathrm{deg}(u), \mathrm{deg}(v)) ,$$ where the summing is performed across all the unordered pairs of adjacent vertices $u$ and $v$, with $\mathcal{F} \colon \mathbb{N} \times \mathbb{N} \to \mathbb{R}$ being a symmetric function such that \mathcal{F}(x, a) + \mathcal{F}(y, b) > \mathcal{F}(y, a) + \mathcal{F}(x, b) \quad \mbox{for any $x > y$ and $a > b$} . In an earlier paper, Wang [Cent. Eur. J. Math. 12 (2014) 1656-1663] demonstrated that the greedy tree must always attain the maximum $R_\mathcal{F}$ value on $\mathcal{T}_D$, while an alternating greedy tree necessarily attains the minimum $R_\mathcal{F}$ value on $\mathcal{T}_D$. In this paper, we implement an algorithmic approach in order to find the full solution set to both the $R_\mathcal{F}$ maximization and $R_\mathcal{F}$ minimization problem on $\mathcal{T}_D$. In other words, we determine all the trees from $\mathcal{T}_D$ that attain the maximum $R_\mathcal{F}$ value on this set, together with all the trees from $\mathcal{T}_D$ that minimize the said graph invariant, thereby improving the aforementioned earlier result obtained by Wang