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.
Highlights
In the context of chemical graph theory, graphs are used to model molecules: the vertices of the graph represent the atoms of the molecule, and the edges correspond to the chemical bonds between atoms
We 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 has a number of applications
The greedy tree G(D) with degree sequence D, formally defined is a tree that can be constructed by starting with the largest degree vertex and always assigning the largest available degree to a neighbour of the vertex with the largest degree whose neighbour degrees are not yet fully specified
Summary
In the context of chemical graph theory, (molecular) graphs are used to model molecules: the vertices of the graph represent the atoms of the molecule, and the edges correspond to the chemical bonds between atoms. Let ρ be an invariant of rooted trees that satisfies the recurrence relation (2). Various special cases are listed at the end of each section This includes many known results, but we obtain several new results, for the number of rooted spanning forests (related to the coefficients of the Laplacian characteristic polynomial), the incidence energy, and an invariant called the solvability. We will see that the greedy tree is the unique ρ-exchange-extremal tree if ρ satisfies the following conditions: I.1 the recurrence relation (2) holds, for some symmetric recurrence rule fρ, I.2 the function fρ is strictly increasing (strictly increasing in each single variable and strictly increasing under addition of further variables), I.3 ρ() < ρ(B), for all rooted trees B with |V (B)| > 1, where denotes a single vertex tree. If T is a ρ0-exchange-extremal tree, T is a greedy tree
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