Operations on Complex Hesitancy Fuzzy Graph

Abstract

In this paper, Complex Hesitancy Fuzzy Graph (CHFG) is analyzed and we have discussed new concepts in CHFG such as Complete CHFG with example. Further we defined some operations on Complex Hesitancy Fuzzy Graph such as direct product, semi strong product and strong product. Also fundamental theorems and their apllications will be examined. Additionally, new ideas in the density of a CHFG and balanced complex Hesitancy fuzzy graph have been introduced in this work.

Introduction

I. INTRODUCTION

Zadeh[19],[20],[21] defined the fuzzy set in 1965, and a membership function with a range of[0,1] is a necessary component. Kalaiarasi and Mahalakshmi[4] provided an Introduction to fuzzy soft graph, fuzzy strong graph and its complement. Some applications of -complement fuzzy soft graph and fuzzy strong graph and complement of fuzzy strong graph. They additionally proposed a colouring of regular and strong arcs of fuzzy graph and they also extended the concept of regular and irregular m-polar fuzzy graph.Then, by incorporating non-belongingness and hesitation degrees into the fuzzy set, Atanassov[2]presented the idea of the intuitionistic fuzzy set in 1986.Atanassov added new components that determine the degree of non-membership.

A innovative idea of hesitant fuzzy sets was refined and generalized in 2009 by Torra and Narukawa[15],[16]. According to the HFS concept, each element or object is assigned a set of independent values via the range of membership values. As a result, as compared to Fuzzy Set and Intuitionistic Fuzzy Set, hesitant sets offer more thorough information than the other uncertainty sets. A hesitant fuzzy element was defined in 2011 by Xia and Xu[18]. Hesitancy Fuzzy Set is a successful device to address decisionmakers' (Dm's) reluctant inclinations. It is involved in grouping, enhancement, convexity, navigation, inclination relations, information mining, and conglomeration administrator. Complex hesitancy fuzzy graph has be discussed by AbuHijleh[1].

Qian[7]and Talafha et al[11] worked on complex hesitancy fuzzy sets and decision making. Veeramani and Suresh[17]developed the operations on complex fuzzy graph. For instance, the Hesitancy Fuzzy data could have various meanings for the same object, but they would be based on different factors or phases. Ramot et al[9],[10] developed complex fuzzy sets, Complex intuitionistic fuzzy sets, and complex reluctant fuzzy sets are utilizing two factors to exhibit the full significance of data to beat the hindrances in dynamic in the numerical model. In hesitant fuzzy form, Ccomplex Hesitancy Fuzzy Set uses two sets of variables to represent uncertainty and periodicity semantics. In hesitant fuzzy form, Complex Hesitancy Fuzzy Set uses two sets of variables to represent uncertainty and periodicity semantics. Numerous researchers have discussed a complex fuzzy set extensively. Alkouri and Salleh [3] developed it into a complex intuitionistic fuzzy set in 2012–2013. Talal et al.[12],[13] recently provided a Complex Hesitancy Fuzzy Set-based generalization of Complex Fuzzy Set, Complex Intuitionistic Fuzzy Set, and Hesitancy Fuzzy Set along with its applications in 2021.This paper consists of three major sections. In section 1 Introduction is described. In section 2,preliminaries are discussed. Complex Hesitancy fuzzy graph, characteristic and complete complex hesitancy fuzzy graph and operations on Complex Hesitancy Fuzzy Set such as Direct Product of Complex Hesitancy Fuzzy Set,Semi Strong Product of Complex Hesitancy Fuzzy Set ,Strong Product of Complex Hesitancy Fuzzy Set,Balanced Complex Hesitancy Fuzzy Set and Density of Complex Hesitancy Fuzzy Set have been introduced.Section 4 is the conclusion of this work.

Conclusion

In this paper, we present the Complex Hesitancy Fuzzy Graph. Also, We had derived the Complex Hesitancy Fuzzy Graph with example. Complex Hesitancy Fuzzy Graph(CHFG) emerge as a sophisticated extension of classical graph theory introducing nuanced approach to modeling uncertainty and imprecation through the integration of Complex Hesitancy Fuzzy Sets. Also we derived the concept of Complex Hesitancy Fuzzy Graph(CHFG) and some theorems on complete complex hesitancy fuzzy graph. Then we have applied some operations on CHFG such as Direct Product, Semi strong product and Strong product. Here balanced and density of a CHFG had been discussed. The above complex hesitancy fuzzy graph is useful for real life applications. We are committed to manage other maintainable improvement objectives for a better world.

References

[1] AbuHijleh, Eman A. \"Complex Hesitant Fuzzy Graph.\" Fuzzy Information and Engineering 15.2 (2023): 149-161. [2] Atanassov. K.T, “Intuitionistic fuzzy sets”, Fuzzy Sets and System,20 (1986),87-97 [3] Alkouri A. U. M andSalleh A. R, Complex atanassov’s intuitionistic fuzzy relation, Abstr. Appl. Anal., vol. 2013, pp. 1–18, 2013. [4] Javaid.M,Kashif.A, and Rashid.T, Hesitant fuzzy graphs and their products, Fuzzy Inf. Eng., vol. 12, no. 2, pp. 238–252, 2020. [5] Kalaiarasi.Kand Mahalakshmi.L, “An introduction to fuzzy strong graph, fuzzy soft graph, complement of fuzzy strong and soft graph”, ISSN 0973-1768 Volume13, Number 6(2017), pp.2235-2254. [6] Karaaslan.F, Hesitant fuzzy graphs and their applications in decision making, J. Intell. Fuzzy Syst., vol. 36, no. 3, pp. 2729–2741, 2019. [7] Qian.G,Wang.H and Feng.X, “Generalized hesitancy fuzzy sets and their application in decision support system”, Knowledge-Based Systems,37,357-365(2013). [8] Rodriguez. R. M, Martinez. L, Torra. V, Xu.Z.S and Herrera.F, “Hesitancy fuzzy sets:State of the art and future directions,” International journal of intelligent systems,29(6),495-524(2014). [9] Ramot D,Milo R,Friedman.M and Kandal.A, “Complex fuzzy sets”,IEEE Trans Fuzzy Syst,vol.10,No.2, pp 171-186,Apr.2002. [10] Ramot D,Milo R,Friedman.M and Kandal.A, “Complex fuzzy logic”,IEEE Trans Fuzzy Syst,vol.11,No.4, pp 450-461,Aug.2003. [11] Talafha.M, Alkouri.A.U ,Alqaraleh.S,Zureigat.H, and Aljarrah.A, Complex hesitant fuzzy sets and its applications in multiple attributes decision making problems, J. Intell. Fuzzy Syst., vol. 41, no. 6, pp. 7299–7327, 2021. [12] Talal AL-Harway, “Complete fuzzy graphs”, International J.Math.Combin. vol.4 (2011),26-34 [13] Talal AL-Harway and Laith Almomani, “Balanced Fuzzy Graphs” Balanced Fuzzy Graphs”, arXiv:1804-08677v1[math.CO]23 Apr 2018. [14] Tamir.D.E Jin.L, and Kandel.A, “ A new interpretation of complex membership grade”, Int. J. Intell. Syst., vol. 26, no. 4, pp. 285– 312, 2011. [15] Torra.V, “Hesitant fuzzy sets,”International Journal of Intelligent Systems” ,vol. 25,no. 6, pp. 529–539, 2010 [16] Torra V,Narukawa Y. “On hesitant fuzzy set and decision”.In:Proc of the 2009 IEEE Int Conf on Fuzzy system,Jeju Island,Korea,Aug 2009. DVD-Rom,pp 1378-1382. [17] Veeramani.V and Suresh.R, “ Characteristics and Operations of Complex Fuzzy Graphs ”,ISSN:2583-5343,Int.J.of IT, Res. & App, Vol.2,No.2,22 June 2023:47-53. [18] Xia.M and Xu.Z, “Hesitant fuzzy intormation aggregation in decition making”,Int.J.Approx.Reason.,vol.52,no.3,pp.395-407,2011. [19] Zadeh, L.A, “ Fuzzy Sets, Information and control”,(8),338-353,1965. [20] Zadeh, L.A, “ Similarity relations and fuzzy ordering, Information science”,3,177-200,1971. [21] Zadeh, L.A, “Is there a need for logic?, Information science”,178,2751-2779,2008. [22] Zeshui Xu, “Hesitant Fuzzy Sets Theory”, Springer International Publishing Switzerland,2014.

Copyright

Copyright © 2024 L Mahalakshmi, M. Sinduja. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.