Trees with the second-minimal ABC energy

Xiaodi Song; Jianping Li; Jianbin Zhang; Weihua He; Xiaodi Song; Jianping Li; Jianbin Zhang; Weihua He

doi:10.3934/math.20221009

AIMS Mathematics

2022, Volume 7, Issue 10: 18323-18333. doi: 10.3934/math.20221009

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Research article

Trees with the second-minimal ABC energy

1.
School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou 510090, China
2.
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
3.
School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an, Shaanxi 710129, China
4.
Xi'an-Budapest Joint Reserch Center for Combinatories, Northwestern Polytechnical University, Xi'an, Shaanxi 710129, China

Received: 30 April 2022 Revised: 21 July 2022 Accepted: 01 August 2022 Published: 15 August 2022
MSC : 05C50

The atom-bond connectivity energy (ABC energy) of an undirected graph $ G $, denoted by $ \mathcal{E}_{ABC}(G) $, is defined as the sum of the absolute values of the ABC eigenvalues of $ G $. Gao and Shao [The minimum ABC energy of trees, Linear Algebra Appl., 577 (2019), 186-203] proved that the star $ S_n $ is the unique tree with minimum ABC energy among all trees on $ n $ vertices. In this paper, we characterize the trees with the minimum ABC energy among all trees on $ n $ vertices except the star $ S_n $.
- energy,
- ABC energy,
- ABC matrix,
- ABC eigenvalues,
- tree
Citation: Xiaodi Song, Jianping Li, Jianbin Zhang, Weihua He. Trees with the second-minimal ABC energy[J]. AIMS Mathematics, 2022, 7(10): 18323-18333. doi: 10.3934/math.20221009

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Abstract

The atom-bond connectivity energy (ABC energy) of an undirected graph $ G $, denoted by $ \mathcal{E}_{ABC}(G) $, is defined as the sum of the absolute values of the ABC eigenvalues of $ G $. Gao and Shao [The minimum ABC energy of trees, Linear Algebra Appl., 577 (2019), 186-203] proved that the star $ S_n $ is the unique tree with minimum ABC energy among all trees on $ n $ vertices. In this paper, we characterize the trees with the minimum ABC energy among all trees on $ n $ vertices except the star $ S_n $.

References

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