Q-borderenergeticity Under the Graph Operation of Complements

Jing Li; Bo Deng; Xumei Jin; Xiaoyun Lv

doi:doi:10.11648/j.acm.20221103.14

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Q-borderenergeticity Under the Graph Operation of Complements

Jing Li, Bo Deng, Xumei Jin, Xiaoyun Lv

Published in Applied and Computational Mathematics (Volume 11, Issue 3)

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 λ_n ≤ λ_n-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(n^1.5). 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 K_n. 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.

Published in	Applied and Computational Mathematics (Volume 11, Issue 3)
DOI	10.11648/j.acm.20221103.14
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.
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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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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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TY  - JOUR
T1  - Q-borderenergeticity Under the Graph Operation of Complements
AU  - Jing Li
AU  - Bo Deng
AU  - Xumei Jin
AU  - Xiaoyun Lv
Y1  - 2022/06/14
PY  - 2022
N1  - https://doi.org/10.11648/j.acm.20221103.14
DO  - 10.11648/j.acm.20221103.14
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 81
EP  - 86
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20221103.14
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
IS  - 3
ER  -

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Author Information

Jing Li

School of Mathematics and Statistics, Qinghai Normal University, Xining, China
Bo Deng

School of Mathematics and Statistics, Qinghai Normal University, Xining, China
Xumei Jin

School of Mathematics and Statistics, Qinghai Normal University, Xining, China
Xiaoyun Lv

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

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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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - Q-borderenergeticity Under the Graph Operation of Complements
AU  - Jing Li
AU  - Bo Deng
AU  - Xumei Jin
AU  - Xiaoyun Lv
Y1  - 2022/06/14
PY  - 2022
N1  - https://doi.org/10.11648/j.acm.20221103.14
DO  - 10.11648/j.acm.20221103.14
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 81
EP  - 86
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20221103.14
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
IS  - 3
ER  -

Copy | Download