Ordering trees by their <scp>ABC</scp> spectral radii

Lin, Wenshui; Yan, Zhangyong; Fu, Peifang; Liu, Jia‐Bao

doi:10.1002/qua.26519

Cited by 5 publications

(5 citation statements)

References 8 publications

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“…adding edge uv. The next result can be used to simplify the calculation of φ(A ex (G); x) in some cases (e.g., trees), which follows from the same arguments as Lemmas 2.1 and 2.2 in [9]; here we omit its proof.…”

Section: Lemma 24 ([7]mentioning

confidence: 97%

On trees with extremal extended spectral radius

Hu¹,

Chen²,

Zhu³

2021

Preprint

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Let G be a simple connected graph with n vertices, and let d i be the degree of the vertex v i in G. The extended adjacency matrix of G is defined so that the ij-entry is 1 2if the vertices v i and v j are adjacent in G, and 0 otherwise. This matrix was originally introduced for developing novel topological indices used in the QSPR/QSAR studies. In this paper, we consider extremal problems of the largest eigenvalue of the extended adjacency matrix (also known as the extended spectral radius) of trees. We show that among all trees of order n ≥ 5, the path P n (resp., the star S n ) uniquely minimizes (resp., maximizes) the extended spectral radius. We also determine the first five trees with the maximal extended spectral radius.

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Section: Lemma 24 ([7]mentioning

confidence: 97%

On trees with extremal extended spectral radius

Hu¹,

Chen²,

Zhu³

2021

Preprint

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“…Estrada [ 33 ] proposed that the atom‐bond connectivity (

) matrix of graphs in the conjugation with ABC index and defined as

Let

be the eigenvalues of ABC are called ABC‐eigenvalues with a graph G . He also proposed the ABC spectral radius and energy [ 34 ] and defined as

Many researchers computed the ABC spectral radius and energy for different graphs [ 35 , 36 , 37 , 38 , 39 , 40 , 41 ].…”

Section: Preliminariesmentioning

confidence: 99%

“…Estrada [33] proposed that the atom‐bond connectivity (

ABC

) matrix of graphs in the conjugation with ABC index and defined as

{(ABC)}_{u, v} goodbreak= ({, \begin{matrix} \sqrt{\frac{d_{u} + d_{v} - 2}{d_{u} d_{v}}} if italicuv \in E ((), G) \\ 0 Otherwise \end{matrix})

Let

λ_{1}^{abc} \geq λ_{2}^{abc} \geq \dots \geq λ_{n}^{abc}

be the eigenvalues of ABC are called ABC‐eigenvalues with a graph G . He also proposed the ABC spectral radius and energy [34] and defined as

ρ_{ABC} goodbreak= ρ_{ABC} ((), G) ≔ λ_{1}^{ABC},

E_{ABC} goodbreak= E_{ABC} ((), G) ≔ \sum_{i = 1}^{n} (||, λ_{i}^{ABC}) .

Many researchers computed the ABC spectral radius and energy for different graphs [35–41].…”

Section: Preliminariesmentioning

confidence: 99%

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Quantitative structure–properties relationship analysis of Eigen‐value‐based indices using COVID‐19 drugs structure

Rauf

Hanif

2022

Int J of Quantum Chemistry

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Topological indices are an important method for understanding the fundamental topology of chemical structures. Quantitative structure properties relationship (QSPR) is an analytical approach for breaking down a molecule into a sequence of numerical values that describe the chemical and physical characteristics of the molecule. In this article, we have developed the QSPR analysis between eigenvalue‐based topological indices and physical properties of COVID‐19 drugs to predict the significance level of eigenvalue based indices. We have to use MATLAB for the computation of indices and SPSS for analysis. We show that positive interia index, signless Laplacian Estrada index and Randić energy are the best predictors of molar reactivity, polar surface area and molecular weight, respectively.

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“…Chen [9] characterized the graphs with extremal ABC spectral radius for a class of given graphs. Lin et al [10] determined the trees with the third, fourth, and fifth largest ABC spectral radii.…”

Section: Introductionmentioning

confidence: 99%

The Maximal ABC Index of the Corona of Two Graphs

Liu

Shao

2021

Mathematical Problems in Engineering

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Let G 1 ∘ G 2 be the corona of two graphs G 1 and G 2 which is the graph obtained by taking one copy of G 1 and V G 1 copies of G 2 and then joining the i th vertex of G 1 to every vertex in the i th copy of G 2 . The atom-bond connectivity index (ABC index) of a graph G is defined as A B C G = ∑ u v ∈ E G d G u + d G v − 2 / d G u d G v , where E G is the edge set of G and d G u and d G v are degrees of vertices u and v , respectively. For the ABC indices of G 1 ∘ G 2 with G 1 and G 2 being connected graphs, we get the following results. (1) Let G 1 and G 2 be connected graphs. The ABC index of G 1 ∘ G 2 attains the maximum value if and only if both G 1 and G 2 are complete graphs. If the ABC index of G 1 ∘ G 2 attains the minimum value, then G 1 and G 2 must be trees. (2) Let T 1 and T 2 be trees. Then, the ABC index of T 1 ∘ T 2 attains the maximum value if and only if T 1 is a path and T 2 is a star.

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Ordering trees by their ABC spectral radii

Cited by 5 publications

References 8 publications

On trees with extremal extended spectral radius

On trees with extremal extended spectral radius

Quantitative structure–properties relationship analysis of Eigen‐value‐based indices using COVID‐19 drugs structure

The Maximal ABC Index of the Corona of Two Graphs

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