American Institute of Mathematical Sciences

Advanced
August & September  2019, 12(4&5): 711-721. doi: 10.3934/dcdss.2019045

Tight independent set neighborhood union condition for fractional critical deleted graphs and ID deleted graphs

Wei Gao 1,, , Darko Dimitrov 2, and  Hosam Abdo 3,

1. 

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

Faculty of Information Studies, Novo Mesto, Slovenia
3. 

Institut für Informatik, Freie Universität Berlin, Takustraße, D-4195 Berlin, Germany

* Corresponding author: W. Gao

Received  July 2017 Revised  December 2017 Published  November 2018

Fund Project: The first author is supported by NSFC grant (no. 11401519)

The problem of data transmission in communication network can betransformed into the problem of fractional factor existing in graph theory. Inrecent years, the data transmission problem in the specificnetwork conditions has received a great deal of attention, and itraises new demands to the corresponding mathematical model. Underthis background, many advanced results are presented on fractionalcritical deleted graphs and fractional ID deleted graphs. In thispaper, we determine that $G$ is a fractional
$ (g,f,n',m) $
-critical deleted graph if
$ δ(G)≥\frac{b^{2}(i-1)}{a}+n'+2m $
,
$ n>\frac{(a+b)(i(a+b)+2m-2)+bn'}{a} $
, and
$|N_{G}(x_{1})\cup N_{G}(x_{2})\cup···\cup N_{G}(x_{i})|≥\frac{b(n+n')}{a+b}$
for any independent subset
$ \{x_{1},x_{2},..., x_{i}\} $
 of
$ V(G) $
. Furthermore, the independent set neighborhood union condition for a graph to be fractional ID-
$ (g,f,m) $
-deleted is raised. Some examples will be manifested to show the sharpness of independent set neighborhood union conditions.
Keywords: Graph, fractional factor, fractional $ (g, f, n', m) $-critical deleted graph, fractional ID-$ (g, f, m) $-deleted graph, independent set neighborhood union.
Mathematics Subject Classification: Primary: 05C70; Secondary: 68M10.
Citation: Wei Gao, 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
References:
[1]

E. I. Abouelmagd and J. L. G. Guirao, On the perturbed restricted three-body problem, Appl. Math. Nonl. Sc., 1 (2016), 123-144. Google Scholar

[2]

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

[3]

R. Y. Chang, G. Z. Liu and Y. Zhu, Degree conditions of fractional ID-$k$-factor-critical graphs, Bull. Malays. Math. Sci. Soc., 33 (2010), 355-360. Google Scholar

[4]

W. Gao, Some Results on Fractional Deleted Graphs, Doctoral disdertation of Soochow university, 2012.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, Y. Guo and K. Y. Wang, Ontology algorithm using singular value decomposition and applied in multidisciplinary, Cluster Comput., 19 (2016), 2201-2210. Google Scholar

[7]

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

[8]

W. Gao and W. F. Wang, The fifth geometric arithmetic index of bridge graph and carbon nanocones, J. Differ. Equ. Appl., 23 (2017), 100-109. doi: 10.1080/10236198.2016.1197214. Google Scholar

[9]

W. Gao and W. F. Wang, The eccentric connectivity polynomial of two classes of nanotubes, Chaos Soliton. Fract., 89 (2016), 290-294. doi: 10.1016/j.chaos.2015.11.035. Google Scholar

[10]

W. Gao, J. L. G. Guirao and H. L. Wu, Two tight independent set conditions for fractional $(g, f, m)$-deleted graphs systems, Qual. Theory Dyn. Syst., 17 (2018), 231-243. doi: 10.1007/s12346-016-0222-z. Google Scholar

[11]

J. L. G. Guirao and A. C. J. Luo, New trends in nonlinear dynamics and chaoticity, Nonlinear Dynam., 84 (2016), 1-2. doi: 10.1007/s11071-016-2656-x. Google Scholar

[12]

S. Z. Zhou, Z. R. Sun and Z. R. Xu, A result on $r$-orthogonal factorizations in digraphs, Eur. J. Combin., 65 (2017), 15-23. doi: 10.1016/j.ejc.2017.05.001. Google Scholar

[13]

S. Z. Zhou, F. Yang and Z. R. Sun, A neighborhood condition for fractional ID-$[a, b]$-factor-critical graphs, Discuss. Mathe. Graph T., 36 (2016), 409-418. doi: 10.7151/dmgt.1864. Google Scholar

[14]

S. Z. Zhou, L. Xu and Y. Xu, A sufficient condition for the existence of a $k$-factor excluding a given $r$-factor, Appl. Math. Nonl. Sc., 2 (2017), 13-20. Google Scholar

[15]

S. Z. Zhou, Some Results about Component Factors in Graphs, RAIRO-Oper. Res., 2017. doi: 10.1051/ro/2017045. Google Scholar

[16]

S. Z. Zhou, Z. R. Sun and H. Liu, A minimum degree condition for fractional ID-[$ a,b $]-factor-critical graphs, Bull. Aust. Math. Soc., 86 (2012), 177-183. doi: 10.1017/S0004972711003467. Google Scholar

show all references

References:
[1]

E. I. Abouelmagd and J. L. G. Guirao, On the perturbed restricted three-body problem, Appl. Math. Nonl. Sc., 1 (2016), 123-144. Google Scholar

[2]

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

[3]

R. Y. Chang, G. Z. Liu and Y. Zhu, Degree conditions of fractional ID-$k$-factor-critical graphs, Bull. Malays. Math. Sci. Soc., 33 (2010), 355-360. Google Scholar

[4]

W. Gao, Some Results on Fractional Deleted Graphs, Doctoral disdertation of Soochow university, 2012.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, Y. Guo and K. Y. Wang, Ontology algorithm using singular value decomposition and applied in multidisciplinary, Cluster Comput., 19 (2016), 2201-2210. Google Scholar

[7]

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

[8]

W. Gao and W. F. Wang, The fifth geometric arithmetic index of bridge graph and carbon nanocones, J. Differ. Equ. Appl., 23 (2017), 100-109. doi: 10.1080/10236198.2016.1197214. Google Scholar

[9]

W. Gao and W. F. Wang, The eccentric connectivity polynomial of two classes of nanotubes, Chaos Soliton. Fract., 89 (2016), 290-294. doi: 10.1016/j.chaos.2015.11.035. Google Scholar

[10]

W. Gao, J. L. G. Guirao and H. L. Wu, Two tight independent set conditions for fractional $(g, f, m)$-deleted graphs systems, Qual. Theory Dyn. Syst., 17 (2018), 231-243. doi: 10.1007/s12346-016-0222-z. Google Scholar

[11]

J. L. G. Guirao and A. C. J. Luo, New trends in nonlinear dynamics and chaoticity, Nonlinear Dynam., 84 (2016), 1-2. doi: 10.1007/s11071-016-2656-x. Google Scholar

[12]

S. Z. Zhou, Z. R. Sun and Z. R. Xu, A result on $r$-orthogonal factorizations in digraphs, Eur. J. Combin., 65 (2017), 15-23. doi: 10.1016/j.ejc.2017.05.001. Google Scholar

[13]

S. Z. Zhou, F. Yang and Z. R. Sun, A neighborhood condition for fractional ID-$[a, b]$-factor-critical graphs, Discuss. Mathe. Graph T., 36 (2016), 409-418. doi: 10.7151/dmgt.1864. Google Scholar

[14]

S. Z. Zhou, L. Xu and Y. Xu, A sufficient condition for the existence of a $k$-factor excluding a given $r$-factor, Appl. Math. Nonl. Sc., 2 (2017), 13-20. Google Scholar

[15]

S. Z. Zhou, Some Results about Component Factors in Graphs, RAIRO-Oper. Res., 2017. doi: 10.1051/ro/2017045. Google Scholar

[16]

S. Z. Zhou, Z. R. Sun and H. Liu, A minimum degree condition for fractional ID-[$ a,b $]-factor-critical graphs, Bull. Aust. Math. Soc., 86 (2012), 177-183. doi: 10.1017/S0004972711003467. Google Scholar

[1]

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
[2]

Philip M. J. Trevelyan. Approximating the large time asymptotic reaction zone solution for fractional order kinetics $A^n B^m$. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 219-234. doi: 10.3934/dcdss.2012.5.219
[3]

Ruiling Tian, Dequan Yue, Wuyi Yue. Optimal balking strategies in an M/G/1 queueing system with a removable server under N-policy. Journal of Industrial & Management Optimization, 2015, 11 (3) : 715-731. doi: 10.3934/jimo.2015.11.715
[4]

Lijia Yan. Some properties of a class of $(F,E)$-$G$ generalized convex functions. Numerical Algebra, Control & Optimization, 2013, 3 (4) : 615-625. doi: 10.3934/naco.2013.3.615
[5]

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

[6]

Oded Schramm. Hyperfinite graph limits. Electronic Research Announcements, 2008, 15: 17-23. doi: 10.3934/era.2008.15.17
[7]

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
[8]

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

[9]

Sheng Zhu, Jinting Wang. Strategic behavior and optimal strategies in an M/G/1 queue with Bernoulli vacations. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1297-1322. doi: 10.3934/jimo.2018008
[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]

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
[12]

Dominique Zosso, Braxton Osting. A minimal surface criterion for graph partitioning. Inverse Problems & Imaging, 2016, 10 (4) : 1149-1180. doi: 10.3934/ipi.2016036
[13]

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
[14]

Yonglin Cao, Yuan Cao, Hai Q. Dinh, Fang-Wei Fu, Jian Gao, Songsak Sriboonchitta. Constacyclic codes of length $np^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$. Advances in Mathematics of Communications, 2018, 12 (2) : 231-262. doi: 10.3934/amc.2018016
[15]

Jerim Kim, Bara Kim, Hwa-Sung Kim. G/M/1 type structure of a risk model with general claim sizes in a Markovian environment. Journal of Industrial & Management Optimization, 2012, 8 (4) : 909-924. doi: 10.3934/jimo.2012.8.909
[16]

Sung-Seok Ko, Jangha Kang, E-Yeon Kwon. An $(s,S)$ inventory model with level-dependent $G/M/1$-Type structure. Journal of Industrial & Management Optimization, 2016, 12 (2) : 609-624. doi: 10.3934/jimo.2016.12.609
[17]

Wenxiang Liu, Thomas Hillen, H. I. Freedman. A mathematical model for M-phase specific chemotherapy including the $G_0$-phase and immunoresponse. Mathematical Biosciences & Engineering, 2007, 4 (2) : 239-259. doi: 10.3934/mbe.2007.4.239
[18]

Chandra Shekhar, Amit Kumar, Shreekant Varshney, Sherif Ibrahim Ammar. $ \bf{M/G/1} $ fault-tolerant machining system with imperfection. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-28. doi: 10.3934/jimo.2019096
[19]

Laurent Imbert, Michael J. Jacobson, Jr.. Empirical optimization of divisor arithmetic on hyperelliptic curves over $\mathbb{F}_{2^m}$. Advances in Mathematics of Communications, 2013, 7 (4) : 485-502. doi: 10.3934/amc.2013.7.485
[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

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (62)
  • HTML views (324)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences