Rough set theory is used in simple directed graphs to study nano topology. Adjacent vertices was used in digraphs only to define their neighborhoods. Four types of neighborhood systems for vertices are introduced in this article which depend on both adjacent vertices and associated edges. Additionally, the generalization of some notions presented by Pawlak and Lellis Thivagar and some of their properties are investigated. Finally, we present a new model of a blood circulation system of the human heart based on blood paths. Also, different kinds of topological separation axioms are presented and studied between vertices and edges of the heart blood circulation model.
AbdM.E., El-Monsef, M. Shokry and Y.Y. Yousif, Near Approximations in Gm closure space, ISRN Applied Mathematics23 (2012), 1–23.
2.
AkramM., Shumaiza and M. Arshad, A new approach based on fuzzy rough digraphs for decision-making, Journal of Intelligent and Fuzzy Systems35(3) (2018), 2105–2121.
3.
AkramM., ArshadM., and Shumaiza, Fuzzy rough graph theory with applications, International Journal of Computational Intelligence Systems12 (2018), 90–107.
4.
BoffeyT.B., Graph theory in operations research, The Macmillan Press LTD, London and Basingstoke, (1982).
5.
BondyJ.A., MurtyU.S.R., Graph Theory with Applications, Elsevier Sience Publishing Co. Inc., (1975).
6.
DiestelR., Graph Theory II, Springer-Verlag, Heidelberg4 (2010).
7.
El-SharkasyM.M., BadrM.S., Topological spaces via phenotype-genotype space, International Journal of Biomathematics9(4) (2016), 1–16.
8.
El-SharkasyM.M., BadrM.S., Modeling DNA and RNA mutation usingset and topology, International Journal of Biomathematics11(4) (2018), 1–16.
9.
El-SharkaysM.M., ShokryM.S., Separation axioms under crossover operator and its generalized, International Journal of Biomathematics9(4) (2016), 1–23.
10.
JiangH., ZhanJ., ChenD., Covering based variable precision (, )-fuzzy rough sets with applications to multi-attribute decision-making, IEEE Transactions on Fuzzy Systems27(8) (2019), 1558–1572.
11.
LashinE.F., KozaeA.M., AboA.A., Khadra and T. Medhat, Rough set theory for topological spaces, Int J Appr Reas40(1-2) (2005), 25–43.
12.
LellisM., Thivagar and C. Richard, On nano forms of weakly open sets, Int J math Stat Invent1 (2013), 31–37.
13.
LellisM., Thivagar, C. Richard and P. Nirmala Rebecca, Mathematical innovations of a modern topology in medical events, Int J Inf Sci2(4) (2012), 33–36.
14.
LellisM., Thivagar and S.P.R. Priyalatha, Medical diagnosis in an indiscernibility matrix based on nano topology, Cogent Math4(1) (2017), 1–9.
15.
MaT., WuJ., HaoL., LiD., Energy flow matrix modeling and optimal operation analysis of multi energy systems based on graph theory, Applied Thermal Engineering146 (2019), 648–663.
16.
NadaS., El AtikA.A. and AtefM., New types of topological structures via graphs, Math Methods Appl Sci41 (2018), 1–10.
17.
NasefA. and El AtikA.A., Some properties on nano topology induced by graphs, AASCIT J Nanosci3(4) (2017), 19–23.
18.
NawarA.S. and El AtikA.A., A model of a human heart via graph nano topological spaces, International Journal of Biomathematics12(1) (2019). DOI: 10.1142/S1793524519500062
19.
PawlakZ., Rough sets, Int J Inf Comput SciII (1982), 341–356.
20.
PawlakZ., Rough Sets, Theoretical Aspects of Reasoning About Data Kluwer Acadmic Publishers, Dordrecht (1991).
21.
SeffensW., Graph theory patterns in the genetic codes, Forma17 (2002), 309–320.
22.
ShokryM. and AlyR.E., Topological properties on graph vs medical application in human heart, Int J Appl Math1(15) (2013), 1103–1109.
23.
SoraggiS. and WiufC., General theory for stochastic admixture graphs and F-statistics, Theoretical Population Biology125 (2019), 56–66.
24.
WeiC-C., et al., Some topological invariants of graphs associated with the group of symmetries, Hindawi, Journal of Chemistry2020 Article ID 6289518, 13 pages. https://doi.org/10.1155/2020/6289518.
ZhanJ., SunB. and AlcantudJ.C.R., Covering based multigranulation (, )-fuzzy rough set models and applications in multi attribute group decision-making, Information Sciences476 (2019), 290–318.
27.
ZhanJ., MalikH.M. and AkramM., Novel decision-making algorithms based on intuitionistic fuzzy rough environment, International Journal of Machine Learning and Cybernetics10 (2019), 1459–1485.
28.
ZhangK., ZhanJ., WuW. and AlcantudJ.C.R., Fuzzy β-covering based (, )-fuzzy rough set models and applications to multi-attribute decision-making, Computers & Industrial Engineering128 (2019), 605–621.
29.
ZhangL., ZhanJ. and XuZ., Covering-based generalized IF rough sets with applications to multi-attribute decision-making, Information Sciences478 (2019), 275–302.
