Extremal trees with fixed degree sequence

TL;DR

This paper presents a general theorem establishing extremal properties of greedy and M-trees for various graph invariants among trees with fixed degree sequences, extending to majorized sequences.

Contribution

It provides a unifying theorem covering many invariants, including new results on rooted forests, incidence energy, and solvability, for trees with fixed or majorized degree sequences.

Findings

Greedy and M-trees are extremal for many invariants.

Theorem applies to invariants like Wiener index, subtrees, independent sets, matchings.

Extensions to majorized degree sequences with multiple applications.

Abstract

The greedy tree $\mathcal{G}(D)$ and the $\mathcal{M}$-tree $\mathcal{M}(D)$ are known to be extremal among trees with degree sequence $D$ with respect to various graph invariants. This paper provides a general theorem that covers a large family of invariants for which $\mathcal{G}(D)$ or $\mathcal{M}(D)$ is extremal. Many known results, for example on the Wiener index, the number of subtrees, the number of independent subsets and the number of matchings follow as corollaries, as do some new results on invariants such as the number of rooted spanning forests, the incidence energy and the solvability. We also extend our results on trees with fixed degree sequence $D$ to the set of trees whose degree sequence is majorised by a given sequence $D$, which also has a number of applications.