On majorization of closed walk vectors of trees with given degree sequences

Chen, Yahong; Gray, Daniel; Jin, Ya-Lei; Zhang, Xiao‐Dong

doi:10.1016/j.amc.2018.05.007

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Lemma 213 ([43]1

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2020

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“…Let T be a tree and S(T ) its subdivision graph. Using the Coulson formula (1) for the energy and the relation (7), we obtain:…”

Section: Lemma 213 ([43]mentioning

confidence: 99%

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Extremal trees with fixed degree sequence

Andriantiana¹,

Misanantenaina²,

Wagner³

2020

Preprint

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The greedy tree G(D) and the M-tree 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 G(D) or 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.

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“…Let T be a tree and S(T ) its subdivision graph. Using the Coulson formula (1) for the energy and the relation (7), we obtain:…”

Section: Lemma 213 ([43]mentioning

confidence: 99%

“…the number of subtrees (and the number of antichains when rooted trees are considered) [3,42], -the spectral radius [5] and the Laplacian spectral radius [40], -the spectral moments [2,24]. See also [7] for a generalisation.…”

Section: Introductionmentioning

confidence: 99%