American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Wireless sensor network energy efficient coverage method based on intelligent optimization algorithm
  • DCDS-S Home
  • This Issue
  • Next Article
    The perturbed photogravitational restricted three-body problem: Analysis of resonant periodic orbits
August & September  2019, 12(4&5): 877-886. doi: 10.3934/dcdss.2019058

An independent set degree condition for fractional critical deleted graphs

Wei Gao 1,, , Juan Luis García Guirao 2, , Mahmoud Abdel-Aty 3,4,5, and  Wenfei Xi 6,

1. 

School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China
2. 

Departamento de Matemática Aplicaday Estadística, Universidad Politécnica de Cartagena, Hospital de Marina, 30203-Cartagena, Región de Murcia, Spain
3. 

Center for Photonics and Smart Materials (CPSM), Zewail City of Science and Technology, Egypt
4. 

Mathematics Department, Faculty of Sciences, Sohag University, Egypt
5. 

Communication and Networks Engineering, Gulf University, Kingdom of Bahrain
6. 

College of Tourism and Geographic Sciences, Yunnan Normal University, Kunming 650500, China

* Corresponding author: Wei Gao(gaowei@ynnu.edu.cn)

Received  November 2017 Revised  January 2018 Published  November 2018

Let $i≥2$, $Δ≥0$, $1≤ a≤ b-Δ$, $n>\frac{(a+b)(ib+2m-2)}{a}+n'$ and $δ(G)≥\frac{b^{2}}{a}+n'+2m$, and let $g,f$ be two integer-valued functions defined on $V(G)$ such that $a≤ g(x)≤ f(x)-Δ≤ b-Δ$ for each $x∈ V(G)$. In this article, it is determined that $G$ is a fractional $(g,f,n',m)$-critical deleted graph if $\max\{d_{1},d_{2},···,d_{i}\}≥\frac{b(n+n')}{a+b}$ for any independent subset $\{x_{1},x_{2},..., x_{i}\}\subseteq V(G)$. The result is tight on independent set degree condition.

Keywords: Graph, fractional factor, fractional critical deleted graph, independent set degree condition.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Wei Gao, Juan Luis García Guirao, Mahmoud Abdel-Aty, Wenfei Xi. An independent set degree condition for fractional critical deleted graphs. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 877-886. doi: 10.3934/dcdss.2019058
References:
[1]

J. A. Bondy and U. S. R. Mutry, Graph Theory, Springer, Berlin, 2008. doi: 10.1007/978-1-84628-970-5.  Google Scholar

[2]

W. Gao, Some Results on Fractional Deleted Graphs, Doctoral disdertation of Soochow university, 2012. Google Scholar

[3]

W. Gao and Y. Gao, Toughness condition for a graph to be a fractional (g, f, n)-critical deleted graph, The Scientific World Jo., 2014 (2014), Article ID 369798, 7 pages, http://dx.doi.org/10.1155/2014/369798. Google Scholar

[4]

W. Gao, L. Liang, T. W. Xu and J. X. Zhou, Tight toughness condition for fractional (g, f, n)-critical graphs, J. Korean Math. Soc., 51 (2014), 55-65.  doi: 10.4134/JKMS.2014.51.1.055.  Google Scholar

[5]

W. Gao, L. Liang, T. W. Xu and J. X. Zhou, Degree conditions for fractional (g, f, n', m)-critical deleted graphs and fractional ID-(g, f, m)-deleted graphs, Bull. Malays. Math. Sci. Soc., 39 (2016), 315-330.  doi: 10.1007/s40840-015-0194-1.  Google Scholar

[6]

W. Gao and M. R. Farahani, Degree-based indices computation for special chemical molecular structures using edge dividing method, Appl. Math. Nonl. Sc., 1 (2016), 94-117.   Google Scholar

[7]

W. Gao and W. F. Wang, Degree conditions for fractional (k, m)-deleted graphs, Ars. Combin., 113A (2014), 273-285.   Google Scholar

[8]

W. Gao and W. F. Wang, Toughness and fractional critical deleted graph, Utilitas Math., 98 (2015), 295-310.   Google Scholar

[9]

W. Gao and W. F. Wang, A tight neighborhood union condition on fractional (g, f, n, m)-critical deleted graphs, Colloq. Math., 149 (2017), 291-298.  doi: 10.4064/cm6959-8-2016.  Google Scholar

[10]

W. Gao and W. F. Wang, New isolated toughness condition for fractional (g, f, n)-critical graphs, Colloq. Math., 147 (2017), 55-65.  doi: 10.4064/cm6713-8-2016.  Google Scholar

[11]

W. Gao and C. C. Yan, A note on fractional (k, n', m)-critical deleted graph, Advances in Computational Mathematics and its Applications, 1 (2012), 53-55.   Google Scholar

[12]

S. Z. Zhou, A minimum degree condition of fractional (k, m)-deleted graphs, Comptes Rendus Math., 347 (2009), 1223-1226.  doi: 10.1016/j.crma.2009.09.022.  Google Scholar

[13]

S. Z. Zhou, A neighborhood condition for graphs to be fractional (k, m)- deleted graphs, Glasg. Math. J., 52 (2010), 33-40.  doi: 10.1017/S0017089509990139.  Google Scholar

[14]

S. Z. Zhou, A sufficient condition for a graph to be a fractional (f, n)-critical graph, Glasgow Math. J., 52 (2010), 409-415.  doi: 10.1017/S001708951000011X.  Google Scholar

[15]

S. Z. Zhou and H. Liu, On fractional (k, m)-deleted graphs with constrains conditions, Int. J. Comput. Math. Sci., 5 (2011), 130-132.   Google Scholar

[16]

S. Z. Zhou, A sufficient condition for graphs to be fractional (k, m)-deleted graphs, Appl. Math. Lett., 24 (2011), 1533-1538.  doi: 10.1016/j.aml.2011.03.041.  Google Scholar

[17]

S. Z. Zhou and Q. X. Bian, An existence theorem on fractional deleted graphs, Period. Math. Hung., 71 (2015), 125-133.  doi: 10.1007/s10998-015-0089-9.  Google Scholar

show all references

References:
[1]

J. A. Bondy and U. S. R. Mutry, Graph Theory, Springer, Berlin, 2008. doi: 10.1007/978-1-84628-970-5.  Google Scholar

[2]

W. Gao, Some Results on Fractional Deleted Graphs, Doctoral disdertation of Soochow university, 2012. Google Scholar

[3]

W. Gao and Y. Gao, Toughness condition for a graph to be a fractional (g, f, n)-critical deleted graph, The Scientific World Jo., 2014 (2014), Article ID 369798, 7 pages, http://dx.doi.org/10.1155/2014/369798. Google Scholar

[4]

W. Gao, L. Liang, T. W. Xu and J. X. Zhou, Tight toughness condition for fractional (g, f, n)-critical graphs, J. Korean Math. Soc., 51 (2014), 55-65.  doi: 10.4134/JKMS.2014.51.1.055.  Google Scholar

[5]

W. Gao, L. Liang, T. W. Xu and J. X. Zhou, Degree conditions for fractional (g, f, n', m)-critical deleted graphs and fractional ID-(g, f, m)-deleted graphs, Bull. Malays. Math. Sci. Soc., 39 (2016), 315-330.  doi: 10.1007/s40840-015-0194-1.  Google Scholar

[6]

W. Gao and M. R. Farahani, Degree-based indices computation for special chemical molecular structures using edge dividing method, Appl. Math. Nonl. Sc., 1 (2016), 94-117.   Google Scholar

[7]

W. Gao and W. F. Wang, Degree conditions for fractional (k, m)-deleted graphs, Ars. Combin., 113A (2014), 273-285.   Google Scholar

[8]

W. Gao and W. F. Wang, Toughness and fractional critical deleted graph, Utilitas Math., 98 (2015), 295-310.   Google Scholar

[9]

W. Gao and W. F. Wang, A tight neighborhood union condition on fractional (g, f, n, m)-critical deleted graphs, Colloq. Math., 149 (2017), 291-298.  doi: 10.4064/cm6959-8-2016.  Google Scholar

[10]

W. Gao and W. F. Wang, New isolated toughness condition for fractional (g, f, n)-critical graphs, Colloq. Math., 147 (2017), 55-65.  doi: 10.4064/cm6713-8-2016.  Google Scholar

[11]

W. Gao and C. C. Yan, A note on fractional (k, n', m)-critical deleted graph, Advances in Computational Mathematics and its Applications, 1 (2012), 53-55.   Google Scholar

[12]

S. Z. Zhou, A minimum degree condition of fractional (k, m)-deleted graphs, Comptes Rendus Math., 347 (2009), 1223-1226.  doi: 10.1016/j.crma.2009.09.022.  Google Scholar

[13]

S. Z. Zhou, A neighborhood condition for graphs to be fractional (k, m)- deleted graphs, Glasg. Math. J., 52 (2010), 33-40.  doi: 10.1017/S0017089509990139.  Google Scholar

[14]

S. Z. Zhou, A sufficient condition for a graph to be a fractional (f, n)-critical graph, Glasgow Math. J., 52 (2010), 409-415.  doi: 10.1017/S001708951000011X.  Google Scholar

[15]

S. Z. Zhou and H. Liu, On fractional (k, m)-deleted graphs with constrains conditions, Int. J. Comput. Math. Sci., 5 (2011), 130-132.   Google Scholar

[16]

S. Z. Zhou, A sufficient condition for graphs to be fractional (k, m)-deleted graphs, Appl. Math. Lett., 24 (2011), 1533-1538.  doi: 10.1016/j.aml.2011.03.041.  Google Scholar

[17]

S. Z. Zhou and Q. X. Bian, An existence theorem on fractional deleted graphs, Period. Math. Hung., 71 (2015), 125-133.  doi: 10.1007/s10998-015-0089-9.  Google Scholar

[1]

Darko Dimitrov, Hosam Abdo. Tight independent set neighborhood union condition for fractional critical deleted graphs and ID deleted graphs. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4-5) : 711-721. doi: 10.3934/dcdss.2019045
[2]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020033
[3]

Shaowen Shi, Weinian Zhang. Bifurcations in an economic model with fractional degree. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020293
[4]

David Auger, Irène Charon, Iiro Honkala, Olivier Hudry, Antoine Lobstein. Edge number, minimum degree, maximum independent set, radius and diameter in twin-free graphs. Advances in Mathematics of Communications, 2009, 3 (1) : 97-114. doi: 10.3934/amc.2009.3.97
[5]

Eric Babson and Dmitry N. Kozlov. Topological obstructions to graph colorings. Electronic Research Announcements, 2003, 9: 61-68.

[6]

J. William Hoffman. Remarks on the zeta function of a graph. Conference Publications, 2003, 2003 (Special) : 413-422. doi: 10.3934/proc.2003.2003.413
[7]

Oded Schramm. Hyperfinite graph limits. Electronic Research Announcements, 2008, 15: 17-23. doi: 10.3934/era.2008.15.17
[8]

John Kieffer and En-hui Yang. Ergodic behavior of graph entropy. Electronic Research Announcements, 1997, 3: 11-16.

[9]

Maoding Zhen, Jinchun He, Haoyun Xu. Critical system involving fractional Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (1) : 237-253. doi: 10.3934/cpaa.2019013
[10]

Roy H. Goodman. NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2203-2232. doi: 10.3934/dcds.2019093
[11]

Christina A. Hollon, Jeffrey T. Neugebauer. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Conference Publications, 2015, 2015 (special) : 615-620. doi: 10.3934/proc.2015.0615
[12]

Mario Roy, Mariusz Urbański. Random graph directed Markov systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 261-298. doi: 10.3934/dcds.2011.30.261
[13]

Dominique Zosso, Braxton Osting. A minimal surface criterion for graph partitioning. Inverse Problems & Imaging, 2016, 10 (4) : 1149-1180. doi: 10.3934/ipi.2016036
[14]

Mario Jorge Dias Carneiro, Rafael O. Ruggiero. On the graph theorem for Lagrangian minimizing tori. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6029-6045. doi: 10.3934/dcds.2018260
[15]

Xiaoming He, Marco Squassina, Wenming Zou. The Nehari manifold for fractional systems involving critical nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1285-1308. doi: 10.3934/cpaa.2016.15.1285
[16]

Vincenzo Ambrosio. Concentration phenomena for critical fractional Schrödinger systems. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2085-2123. doi: 10.3934/cpaa.2018099
[17]

Hui Zhang, Jun Wang, Fubao Zhang. Semiclassical states for fractional Choquard equations with critical growth. Communications on Pure & Applied Analysis, 2019, 18 (1) : 519-538. doi: 10.3934/cpaa.2019026
[18]

Chun-Xiang Guo, Guo Qiang, Jin Mao-Zhu, Zhihan Lv. Dynamic systems based on preference graph and distance. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1139-1154. doi: 10.3934/dcdss.2015.8.1139
[19]

Liu Hui, Lin Zhi, Waqas Ahmad. Network(graph) data research in the coordinate system. Mathematical Foundations of Computing, 2018, 1 (1) : 1-10. doi: 10.3934/mfc.2018001
[20]

Mirela Domijan, Markus Kirkilionis. Graph theory and qualitative analysis of reaction networks. Networks & Heterogeneous Media, 2008, 3 (2) : 295-322. doi: 10.3934/nhm.2008.3.295

2019 Impact Factor: 1.233

Tools

Article outline

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences