Let A(G) be the adjacency matrix of graph G. Suppose λ_{n} ≤ λ_{n-1} ≤ ··· ≤λ_{1} 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.
DOI | 10.11648/j.acm.20221103.14 |
Published in | Applied and Computational Mathematics (Volume 11, Issue 3, June 2022) |
Page(s) | 81-86 |
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Signless Laplacian Energy, Q-borderenergetic Graphs, Zagreb Index, Complement
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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
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
@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} }
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 -