Applied and Computational Mathematics

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Q-borderenergeticity Under the Graph Operation of Complements

Received: 20 May 2022    Accepted: 2 June 2022    Published: 14 June 2022
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Abstract

Let A(G) be the adjacency matrix of graph G. Suppose λn ≤ λn-1 ≤ ··· ≤λ1 are the eigenvalues of A(G). The energy of a graph G is denoted by ε(G), which is defined as the sum of absolute values of its eigenvalues. It is well known that graph energy is found that there are many applications in chemistry. Nikiforov showed that almost all graphs have an energy asymptotically equal to O(n1.5). So, almost all graphs are supperenergetic, i.e., their graph energies are more than those of complete graphs with the same orders. This made an end to the study of supperenergetic graphs. Then the concept of a borderenergetic graph is proposed by Gutman et al. in 2015. If a graph G of order n satisfies it energy ε(G)=2(n-1), then G is called a borderenergetic graph. Recently, Tao and Hou extend this concept to signless Laplacian energy. That is, a graph of order n is called Q-borderenergetic graph if its signless Laplacian energy is equal to that of the complete graph Kn. In this work, by using the graph operation of complements, we find that, for most of Q-borderenergetic graphs, it can not satisfy themselves and their complements are all Q-borderenergetic. Besides, a new lower bound on signless Laplacian energy of the complement of a Q-borderenergetic graph is established.

DOI 10.11648/j.acm.20221103.14
Published in Applied and Computational Mathematics (Volume 11, Issue 3, June 2022)
Page(s) 81-86
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Signless Laplacian Energy, Q-borderenergetic Graphs, Zagreb Index, Complement

References
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[2] J. A. Bondy, U. S. R. Murty, Graph Theory. GTM 244, Springer, Berlin, Germany, 2008.
[3] K. C. Das, S. A. Mojallal, Upper Bounds for the Energy of Graphs. MAT-CH Commun. Math. Comput. Chem. 70 (2013) 657-662.
[4] K. C. Das, S. A. Mojallal, Relation between Energy and (Signless) Laplaci-an Energy of Graphs. MATCH Commun. Math. Comput. Chem. 74 (2015) 359-36.
[5] K. C. Das, S. A. Mojallal, On Laplacian energy of graphs. Discrete Mathe-matics. 325 (2014) 52-64.
[6] B. Deng, C. Chang, H. Zhao, K. C. Das, Construction for the Sequences of Q-borderenergetic Graphs. Mathematical Problems in Engineering. https://doi.org/10.1155/2020/6176849 (2020).
[7] B. Deng, X. Li, More on L-borderenergetic graphs. MATCH Commun. Mat-h. Comput. Chem. 77 (2017) 115–127.
[8] B. Deng, X. Li, Energies for the complements of borderenergetic graphs. MATCH Commun. Math. Comput. Chem. 85 (2021) 181–194.
[9] B. Deng, X. Li, I. Gutman, More on borderenergetic graphs. Linear Al-geb-ra and its Applications. 497 (2016) 199-208.
[10] B. Deng, X. Li, Y. Li, (Signless) Laplacian borderenergetic graphs and th-e join of graphs. MATCH Commun. Math. Comput. Chem. 80 (2018) 449–457.
[11] B. Deng, X. Li, J. Wang, Further results on L-borderenergetic graphs. MA-TCH Commun. Math. Comput. Chem. 77 (2017) 607-616.
[12] B. Deng, X. Li, H. Zhao, (Laplacian) borderenergetic graphs and bipartite graphs. MATCH Commun. Math. Comput. Chem. 82 (2019) 481–489.
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[15] I. Gutman, E. Gudiño, D. Quiroz, Upper bound for the energy of graphs with fixed second and fourth spectral moments. Kragujevac Journal of Mat-hematics. 32 (2009) 27-35.
[16] I. Gutman, The energy of a graph. Discrete Applied Mathematics. 160 (2012) 2177-2187.
[17] S. C. Gong, X. Li, G. H. Xu, I. Gutman, B. Furtula, Borderenergetic gra-phs. MATCH Commun. Math. Comput. Chem. 74 (2015) 321–332.
[18] X. Li, Y. Shi, I. Gutman, Graph Energy. Springer, New York, 2012.
[19] X. Li, M. Wei, S. Gong, A Computer Search for the Borderenergetic Grap-hs of Order 10. MATCH Commun. Math. Comput. Chem. 74 (2015) 333-342.
[20] Huiqing. Liu, M. Lu, F. Tian, Some upper bounds for the energy of graph-s. Journal of Mathematical Chemistry. 41 (2007) 45-57.
[21] X. Lv, B. Deng, X. Li, Laplacian borderenergetic graphs and their comple-ments. MATCH Commun. Math. Comput. Chem. 86 (2021) 587–596.
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Cite This Article
  • APA Style

    Jing Li, Bo Deng, Xumei Jin, Xiaoyun Lv. (2022). Q-borderenergeticity Under the Graph Operation of Complements. Applied and Computational Mathematics, 11(3), 81-86. https://doi.org/10.11648/j.acm.20221103.14

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    ACS Style

    Jing Li; Bo Deng; Xumei Jin; Xiaoyun Lv. Q-borderenergeticity Under the Graph Operation of Complements. Appl. Comput. Math. 2022, 11(3), 81-86. doi: 10.11648/j.acm.20221103.14

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    AMA Style

    Jing Li, Bo Deng, Xumei Jin, Xiaoyun Lv. Q-borderenergeticity Under the Graph Operation of Complements. Appl Comput Math. 2022;11(3):81-86. doi: 10.11648/j.acm.20221103.14

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  • @article{10.11648/j.acm.20221103.14,
      author = {Jing Li and Bo Deng and Xumei Jin and Xiaoyun Lv},
      title = {Q-borderenergeticity Under the Graph Operation of Complements},
      journal = {Applied and Computational Mathematics},
      volume = {11},
      number = {3},
      pages = {81-86},
      doi = {10.11648/j.acm.20221103.14},
      url = {https://doi.org/10.11648/j.acm.20221103.14},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20221103.14},
      abstract = {Let A(G) be the adjacency matrix of graph G. Suppose λn ≤ λn-1 ≤ ··· ≤λ1 are the eigenvalues of A(G). The energy of a graph G is denoted by ε(G), which is defined as the sum of absolute values of its eigenvalues. It is well known that graph energy is found that there are many applications in chemistry. Nikiforov showed that almost all graphs have an energy asymptotically equal to O(n1.5). So, almost all graphs are supperenergetic, i.e., their graph energies are more than those of complete graphs with the same orders. This made an end to the study of supperenergetic graphs. Then the concept of a borderenergetic graph is proposed by Gutman et al. in 2015. If a graph G of order n satisfies it energy ε(G)=2(n-1), then G is called a borderenergetic graph. Recently, Tao and Hou extend this concept to signless Laplacian energy. That is, a graph of order n is called Q-borderenergetic graph if its signless Laplacian energy is equal to that of the complete graph Kn. In this work, by using the graph operation of complements, we find that, for most of Q-borderenergetic graphs, it can not satisfy themselves and their complements are all Q-borderenergetic. Besides, a new lower bound on signless Laplacian energy of the complement of a Q-borderenergetic graph is established.},
     year = {2022}
    }
    

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    AU  - Jing Li
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    AB  - Let A(G) be the adjacency matrix of graph G. Suppose λn ≤ λn-1 ≤ ··· ≤λ1 are the eigenvalues of A(G). The energy of a graph G is denoted by ε(G), which is defined as the sum of absolute values of its eigenvalues. It is well known that graph energy is found that there are many applications in chemistry. Nikiforov showed that almost all graphs have an energy asymptotically equal to O(n1.5). So, almost all graphs are supperenergetic, i.e., their graph energies are more than those of complete graphs with the same orders. This made an end to the study of supperenergetic graphs. Then the concept of a borderenergetic graph is proposed by Gutman et al. in 2015. If a graph G of order n satisfies it energy ε(G)=2(n-1), then G is called a borderenergetic graph. Recently, Tao and Hou extend this concept to signless Laplacian energy. That is, a graph of order n is called Q-borderenergetic graph if its signless Laplacian energy is equal to that of the complete graph Kn. In this work, by using the graph operation of complements, we find that, for most of Q-borderenergetic graphs, it can not satisfy themselves and their complements are all Q-borderenergetic. Besides, a new lower bound on signless Laplacian energy of the complement of a Q-borderenergetic graph is established.
    VL  - 11
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    ER  - 

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Author Information
  • School of Mathematics and Statistics, Qinghai Normal University, Xining, China

  • School of Mathematics and Statistics, Qinghai Normal University, Xining, China; Academy of Plateau Science and Sustainability, Xining, China; The State Key Laboratory of Tibetan Intelligent Information Processing and Application, Xining, China

  • School of Mathematics and Statistics, Qinghai Normal University, Xining, China

  • School of Mathematics and Statistics, Qinghai Normal University, Xining, China

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