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The perturbed photogravitational restricted three-body problem: Analysis of resonant periodic orbits
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 |
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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.
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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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