Extremal Trees with Fixed Degree Sequence

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.

Note on Sombor index of connected graphs with given degree sequence

“…In what follows, we suppose that x 1 ≥ y 1 ≥ 1 and x 2 ≥ y 2 ≥ 0. It suffices to show that h(x, y) satisfies (2). Since the equality holds in (2) for either x 1 = y 1 or…”

“…a certain graph invariant. Several such researches have been published [2,11,13,14,19]. Among these results, in many cases the extremal graphs are BF S-type (BF S = breath first search).…”