# American Institute of Mathematical Sciences

• 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

 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.

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, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033 [3] Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks & Heterogeneous Media, 2021, 16 (2) : 155-185. doi: 10.3934/nhm.2021003 [4] Shaowen Shi, Weinian Zhang. Bifurcations in an economic model with fractional degree. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4407-4431. doi: 10.3934/dcdsb.2020293 [5] 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 [6] Eric Babson and Dmitry N. Kozlov. Topological obstructions to graph colorings. Electronic Research Announcements, 2003, 9: 61-68. [7] Oded Schramm. Hyperfinite graph limits. Electronic Research Announcements, 2008, 15: 17-23. doi: 10.3934/era.2008.15.17 [8] 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 [9] John Kieffer and En-hui Yang. Ergodic behavior of graph entropy. Electronic Research Announcements, 1997, 3: 11-16. [10] Roberto De Leo, James A. Yorke. The graph of the logistic map is a tower. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5243-5269. doi: 10.3934/dcds.2021075 [11] Claudianor O. Alves, César T. Ledesma. Multiplicity of solutions for a class of fractional elliptic problems with critical exponential growth and nonlocal Neumann condition. Communications on Pure & Applied Analysis, 2021, 20 (5) : 2065-2100. doi: 10.3934/cpaa.2021058 [12] Roy H. Goodman. NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 2203-2232. doi: 10.3934/dcds.2019093 [13] Mario Roy, Mariusz Urbański. Random graph directed Markov systems. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 261-298. doi: 10.3934/dcds.2011.30.261 [14] Rui Wang, Rundong Zhao, Emily Ribando-Gros, Jiahui Chen, Yiying Tong, Guo-Wei Wei. HERMES: Persistent spectral graph software. Foundations of Data Science, 2021, 3 (1) : 67-97. doi: 10.3934/fods.2021006 [15] Dominique Zosso, Braxton Osting. A minimal surface criterion for graph partitioning. Inverse Problems & Imaging, 2016, 10 (4) : 1149-1180. doi: 10.3934/ipi.2016036 [16] Mario Jorge Dias Carneiro, Rafael O. Ruggiero. On the graph theorem for Lagrangian minimizing tori. Discrete & Continuous Dynamical Systems, 2018, 38 (12) : 6029-6045. doi: 10.3934/dcds.2018260 [17] 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 [18] 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 [19] Gökhan Mutlu. On the quotient quantum graph with respect to the regular representation. Communications on Pure & Applied Analysis, 2021, 20 (2) : 885-902. doi: 10.3934/cpaa.2020295 [20] 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

2020 Impact Factor: 2.425