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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. [2] W. Gao, Some Results on Fractional Deleted Graphs, Doctoral disdertation of Soochow university, 2012. [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. [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. [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. [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. [7] W. Gao and W. F. Wang, Degree conditions for fractional (k, m)-deleted graphs, Ars. Combin., 113A (2014), 273-285. [8] W. Gao and W. F. Wang, Toughness and fractional critical deleted graph, Utilitas Math., 98 (2015), 295-310. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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. [2] W. Gao, Some Results on Fractional Deleted Graphs, Doctoral disdertation of Soochow university, 2012. [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. [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. [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. [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. [7] W. Gao and W. F. Wang, Degree conditions for fractional (k, m)-deleted graphs, Ars. Combin., 113A (2014), 273-285. [8] W. Gao and W. F. Wang, Toughness and fractional critical deleted graph, Utilitas Math., 98 (2015), 295-310. [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. [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. [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. [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. [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. [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. [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. [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. [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.
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