Laplacian coefficient, matching polynomial and incidence energy of trees with described maximum degree

Jin, Ya-Lei; Yeh, Yeong‐Nan; Zhang, Xiaodong

doi:10.1007/s10878-015-9977-4

Cited by 8 publications

(7 citation statements)

References 18 publications

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“…We show that the invariant rf(•, x) fits our general scheme. This generalises the approach taken in [20], where trees with given maximum degree are considered.…”

Section: Rooted Spanning Forests Incidence Energy Laplacian-energy-mentioning

confidence: 86%

“…Again, several extremality results now follow as direct applications of Corollary 41. For example, the Volkmann tree minimises the Wiener index [10] and the polynomial rf(T, x) for every x > 0 [20] (thus also the incidence energy), while it maximises the number of subtrees [23]. On the other hand, the M-tree M(B) is known to be extremal for the Hosoya index, the energy of a graph and the Merrifield-Simmons index [18,16].…”

Section: Solvabilitymentioning

confidence: 99%

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Extremal Trees with Fixed Degree Sequence

Andriantiana¹,

Misanantenaina

Wagner

2021

Electron. J. Combin.

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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.

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“…We show that the invariant rf(•, x) fits our general scheme. This generalises the approach taken in [20], where trees with given maximum degree are considered.…”

Section: Rooted Spanning Forests Incidence Energy Laplacian-energy-mentioning

confidence: 86%

Section: Solvabilitymentioning

confidence: 99%

Extremal Trees with Fixed Degree Sequence

Andriantiana¹,

Misanantenaina

Wagner

2021

Electron. J. Combin.

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“…By Lemma 2.6, m j (U 2 −2,2t−2s ) ≥ m j (U 2 −1,2t−2s−1 ), where 0 ≤ j ≤ k − 2. Therefore, by (23) and (24), we obtain…”

Section: Preliminariesmentioning

confidence: 99%

“…Among all the trees with a given matching number, Ilić [22] obtained the graph with the minimum IE. Among all the trees with a described maximum degree, Jin et al [23] characterized the trees with the minimum IE. Among all the trees with a given pendent vertex number, Zhang et al [24] got the one with the minimum IE.…”

Section: Introductionmentioning

confidence: 99%

The signless Laplacian coefficients and the incidence energy of unicyclic graphs with given pendent vertices

Wang¹,

Zhong²

2019

Filomat

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Let U r n be the set of unicyclic graphs with n vertices and r pendent vertices (namely, r leaves), where n ≥ 4 and r ≥ 1. We consider the signless Laplacian coefficients (SLCs) and the incidence energy (IE) in U r n. Firstly, among a subset of U r n in which each graph has a fixed odd girth ≥ 3, where n ≥ + 1 and r ≥ 1, we characterize a unique extremal graph which has the minimum SLCs and the minimum IE. Secondly, if G ∈ U r n and G has odd girth ≥ 5, where n ≥ 7 and r ≥ 1, then we prove that a unique extremal graph L n ∈ U r n with girth 4 satisfies that both the SLCs and the IE of G are more than the counterparts of L n .

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“…For the graphs with the extremal IEs and the upper and lower bounds of IE, one can refer to Refs. [5,7,10,15,20,21,24,26,27]. Kaya and Maden got some bounds for the generalized version of incidence energy [12].…”

Section: Introductionmentioning

confidence: 99%

The signless Laplacian coefficients and the incidence energy of graphs with a given bipartition

Zhong

Wang

2020

Filomat

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We consider two classes of the graphs with a given bipartition. One is trees and the other is unicyclic graphs. The signless Laplacian coefficients and the incidence energy are investigated for the sets of trees/unicyclic graphs with n vertices in which each tree/unicyclic graph has an (n1,n2)-bipartition, where n1 and n2 are positive integers not less than 2 and n1+n2 = n. Four new graph transformations are proposed for studying the signless Laplacian coefficients. Among the sets of trees/unicyclic graphs considered, we obtain exactly, for each, the minimal element with respect to the quasi-ordering according to their signless Laplacian coefficients and the element with the minimal incidence energies.

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Laplacian coefficient, matching polynomial and incidence energy of trees with described maximum degree

Cited by 8 publications

References 18 publications

Extremal Trees with Fixed Degree Sequence

Extremal Trees with Fixed Degree Sequence

The signless Laplacian coefficients and the incidence energy of unicyclic graphs with given pendent vertices

The signless Laplacian coefficients and the incidence energy of graphs with a given bipartition

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