In collaboration with Payame Noor University and the Iranian Society of Instrumentation and Control Engineers

Document Type : Research Article

Authors

Department of Mathematics‎, ‎Payame Noor University (PNU), ‎P.O‎. ‎Box‎. ‎19395-4697‎, ‎Tehran‎, ‎Iran

Abstract

In this research‎, ‎we use averages and relative measures of interval grey numbers to introduce grey vertices, ‎grey edges‎, ‎and grey graphs (graphs are based on interval grey numbers)‎. ‎To do so‎, ‎we design a grey graph based on a graph (as the underlying graph)‎. ‎Also‎, ‎we find a relation between grey vertices and grey edges of a grey graph‎. ‎The primary method used in this research is based on linear inequalities related to grey vertices and grey edges‎. ‎We find some necessary and sufficient conditions on the grey vertex (as (non-)discrete grey vertices) connectivity of grey graphs based on interval grey numbers and linear inequality systems}.‎The paper includes implications for the development of(non-)weighted graphs‎, ‎and the modeling of uncertainty problems by grey vertices‎, ‎grey edges‎, ‎and their relations in a grey model as a grey graph‎. ‎As a weighted graph‎, ‎a fuzzy graph is a vital graph that has some applications in the real world‎, ‎but with changes in conditions‎, ‎it loses its efficiency‎. ‎On the other hand‎, ‎the efficiency of a grey graph is stable under changes in the conditions‎. ‎So‎, ‎grey graphs cover the weaknesses of fuzzy graphs‎. ‎The new conception of grey graphs based on grey numbers is introduced in this study‎. ‎We propose an optimization method that can be applied for grey numbers in an extension of graphs‎, ‎and apply it for gray numbers in the real world, especially for optimization problems and via gray graphs.

Keywords

[1] Chen N., Xie N. (2020). “Uncertainty representation and information measurement of grey numbers”, Grey Systems: Theory and Applications, 10(4), 495-512.
[2] Cui J., Tan Q., Zhang C., Yang B. (2021). “A novel framework of graph Bayesian optimization and its applications to real-world network analysis”, Expert Systems with Applications, 170(15), 114524.
[3] Darvishi Salookolaei D., Forrest J., Liu S. (2019). “A comparative analysis of grey ranking approaches”, Grey Systems: Theory and Applications, 9(4), 472-487.
[4] Darvishi Salookolaei D., Forrest J., Liu S. (2021). “Grey linear programming: a survey on solving approaches and applications”, Grey Systems: Theory and Applications, 11(1), 110-135.
[5] Darvishi Salookolaei D., Liu S., Babaei P. (2017). “Application of grey system theory in rainfall estimation”, Control and Optimization in Applied Mathematics, 2(2), 15-31.
[6] Deng J. (1989). “Introduction to grey system theory”, Journal of Grey Systems, 1(1), 1-24.
[7] Jalving J., Cao Y., Zavala V.M. (2019) .“Graph-based modeling and simulation of complex systems”, Computers & Chemical Engineering, 125(9), 134-154.
[8] Khatamov N.M. (2020). “Translation-invariant extreme Gibbs measures for the Blume-Capel model with wand on a Cayley tree”, Ukrainian Mathematical Journal, 72, 623-641.
[9] Liab L., Wang P., Yan J., Wang Y., Lib S., Jiang J., Sun Z., Tang B., Chang T.H., Wang S., Liu Y. (2020). “Real-world data medical knowledge graph: construction and applications”, Artificial Intelligence in Medicine, 103, 101817.
[10] Lin Y., Chen M.Y., Liu S.F. (2004). “Theory of grey systems: capturing uncertainties of grey information”, Kybernetes, 33(2), 196-218.
[11] Liu S., Lin Y. (2006). “Grey Information, theory and practical applications”, Springer-Verlag, London.
[12] Luo D., Huihui Z. (2020). “Grey clustering model based on Kernel and information field”, Grey Systems: Theory and Applications, 10(1), 56-67.
[13] Mierzwiak R., Nowak M., Xie N. (2020). “A new approach to the degree of greyness”, Grey Systems: Theory and Applications, 11(2), 241-252.
[14] Mordeson J.N., Nair P.S. (2000). “Fuzzy graph and fuzzy hypergraph”, Physica-Verlag, Heidelberg, New York.
[15] Pourofoghi F., Darvishi Salookolaei D. (2020). “Applying Duality Results to Solve the Linear Programming Problems with Grey Parameters”, Control and Optimization in Applied Mathematics, 5(1), 15-28.
[16] Tavakolli M. ( 2019). “Global Forcing Number for Maximal Matchings under Graph Operations”, Control and Optimization in Applied Mathematics, 4(1), 53-63.
[17] Ward J.P., Narcowich F.J., Ward J.D. (2020). “Interpolating splines on graphs for data science applications”, Applied and Computational Harmonic Analysis, 49(2), 540-557.
[18] Yang Y., John R. (2012). “Grey sets and greyness”, Information Sciences, 185(1), 249-264.
[19] Yin K., Xu J., Li X. (2019). “A new grey comprehensive relational model based on weighted mean distance and induced intensity and its application”, Grey Systems: Theory and Applications, 9(3), 374-384.
[20] Yin K., Xu T., Li X., Cao Y. (2021). “A study of the grey relational model of interval numbers for panel data”, Grey Systems: Theory and Applications, 11(1), 200-211.
[21] Zadeh L. (1965). “Fuzzy sets ”, Information and Control, 8(3), 338-353.