August  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

1. 

Faculty of Information Studies, Novo Mesto, Slovenia

2. 

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

Received  July 2017 Revised  December 2017 Published  November 2018

Addendum: Wei Guo was withdraw from the author list for the article
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.
Citation: 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. ChangG. 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. GaoL. LiangT. 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. GaoY. 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. GaoL. LiangT. 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. GaoJ. 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. ZhouZ. 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. ZhouF. 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. ZhouL. 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. ZhouZ. 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. ChangG. 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. GaoL. LiangT. 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. GaoY. 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. GaoL. LiangT. 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. GaoJ. 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. ZhouZ. 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. ZhouF. 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. ZhouL. 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. ZhouZ. 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]

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

[2]

W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349

[3]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088

[4]

Khosro Sayevand, Valeyollah Moradi. A robust computational framework for analyzing fractional dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021022

[5]

Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021021

[6]

Zhimin Chen, Kaihui Liu, Xiuxiang Liu. Evaluating vaccination effectiveness of group-specific fractional-dose strategies. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021062

[7]

Huy Dinh, Harbir Antil, Yanlai Chen, Elena Cherkaev, Akil Narayan. Model reduction for fractional elliptic problems using Kato's formula. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021004

[8]

Liangliang Ma. Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021068

[9]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

[10]

Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113

[11]

Dayalal Suthar, Sunil Dutt Purohit, Haile Habenom, Jagdev Singh. Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021019

[12]

Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021017

[13]

Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021020

[14]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023

[15]

Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201

[16]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109

[17]

Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243

[18]

Fernando P. da Costa, João T. Pinto, Rafael Sasportes. On the convergence to critical scaling profiles in submonolayer deposition models. Kinetic & Related Models, 2018, 11 (6) : 1359-1376. doi: 10.3934/krm.2018053

[19]

Gioconda Moscariello, Antonia Passarelli di Napoli, Carlo Sbordone. Planar ACL-homeomorphisms : Critical points of their components. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1391-1397. doi: 10.3934/cpaa.2010.9.1391

[20]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912

2019 Impact Factor: 1.233

Article outline

[Back to Top]