Computer search for large trees with minimal ABC index

Lin, Wenshui; Chen, Jianfeng; Wu, Zhixi; Dimitrov, Darko; Huang, Linshan

doi:10.1016/j.amc.2018.06.012

“…Hence a much more efficient search algorithm or implementation is still desired to identify large min-ABC trees. From Theorem 4.1 the features of an optimal tree degree sequence obtained in [22] and [23] were significantly refined in [11] as following.…”

Section: Greedy Trees and Search Based On Tree Degree Sequencementioning

confidence: 99%

Characterizing Trees with Minimal ABC Index with Computer Search: A Short Survey

Zheng¹,

Lin²,

Qian³

et al. 2018

Open J. Discret. Appl. Math.

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The atom-bond connectivity (ABC ) index of a graph G = (V, E) is defined as ABC(G), where d i denotes the degree of vertex v i of G. Due to its interesting applications in chemistry, this molecular structure descriptor has become one of the most actively studied vertex-degree-based graph invariants. Many efforts were made towards the elementary problem of characterizing tree(s) with minimal ABC index, which remains open and was coined as the "ABC index conundrum". Up to date, quite a few significant results have been obtained. In the course of research computer search plays a non-negligible role. In the present paper we review the state of the art of the problem. In addition we intend to demonstrate that, repeating the procedure "searching -conjecturing -proving" can be an applicable paradigm to cope with elusive problems of extremal graph characterization.

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“…Since the number of n-vertex trees increases rapidly with n, though a computer grid with 400 CPUs was employed, the computation was just performed up to n = 31. One can refer to [11] for the search results. Few as the search results obtained in [10] are, some structural properties of min-ABC trees were observed and proved by Gutman et al in [11].…”

Section: A Brute-force and A Heuristic Search And The Modulo 7 Conjecturementioning

confidence: 99%

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2018

Open J. Discret. Appl. Math.

0

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The atom-bond connectivity (ABC ) index of a graph G = (V, E) is defined as ABC(G), where d i denotes the degree of vertex v i of G. Due to its interesting applications in chemistry, this molecular structure descriptor has become one of the most actively studied vertex-degree-based graph invariants. Many efforts were made towards the elementary problem of characterizing tree(s) with minimal ABC index, which remains open and was coined as the "ABC index conundrum". Up to date, quite a few significant results have been obtained. In the course of research computer search plays a non-negligible role. In the present paper we review the state of the art of the problem. In addition we intend to demonstrate that, repeating the procedure "searching -conjecturing -proving" can be an applicable paradigm to cope with elusive problems of extremal graph characterization.

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“…This topological index turns out to be closely correlated with the heat of formation of alkanes, and became a hot topic in the last few years (see refs. [2–4] and the references therein). In 2017, Estrada [ 5 ] introduced the ABC matrix of G as M ( G ) = ( m ij ) n × n , where m ij = f ( d i , d j ) if v i v j ∈ E , and 0 otherwise.…”

Section: Introductionmentioning

confidence: 99%

Ordering trees by their ABC spectral radii

Lin

¹

,

Yan

²

,

Fu

³

et al. 2020

Int J of Quantum Chemistry

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Let G = (V, E) be a connected graph, where V = {v1, v2, …, vn}. Let di denote the degree of vertex vi. The ABC matrix of G is defined as M(G) = (mij)n × n, where if vivj ∈ E, and 0 otherwise. The ABC spectral radius of G is the largest eigenvalue of M(G). In the present paper, two graph perturbations with respect to ABC spectral radius are established. By applying these perturbations, the trees with the third, fourth, and fifth largest ABC spectral radii are determined.

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Computer search for large trees with minimal ABC index

Cited by 17 publications

References 12 publications

Characterizing Trees with Minimal ABC Index with Computer Search: A Short Survey

Characterizing Trees with Minimal ABC Index with Computer Search: A Short Survey

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Ordering trees by their ABC spectral radii

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