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In this study, a novel concept of picture fuzzy threshold graph (PFTG) is introduced. It has been shown that PFTGs are free from alternating 4-cycle and it can be constructed by repeatedly adding a dominating or an isolated node. Several properties about PFTGs are discussed and obtained the results that every picture fuzzy graph (PFG) is equivalent to a PFTG under certain conditions. Also, the underlying crisp graph (UCG) of PFTG is a split graph (SG), and conversely, a given SG can be applied to constitute a PFTG. A PFTG can be decomposed in a unique way and it generates three distinct fuzzy threshold graphs (FTGs). Furthermore, two important parameters i.e., picture fuzzy (PF) threshold dimension (TD) and PF partition number (PN) of PFGs are defined. Several properties on TD and PN have also been discussed. Lastly, an application of these developed results are presented in controlling medicine resources.

Fuzzy mixed graph (FMG) can be used to model some graphical intercommunication network systems if there are many directed and undirected relations between some vertices. In inverse fuzzy graph, the membership value of edges are greater than or equal to the minimum of the membership value of the corresponding vertices. In inverse fuzzy mixed graph (IFMG), directed and undirected relations exist between some vertices and it can be used to analyze many graphical problems of real life such that the membership values of edges are greater than or equal to the minimum of the membership value of the corresponding pair of vertices. In this article, the concept of IFMG is introduced first with some of its properties. Then some isomorphic properties are studied and complement of IFMG is given. Different types of operations like union, intersection, product and join between two IFMGs are defined and investigated some of their related results. An algorithm of the proposed method is executed to identify some vertices. An application is depicted using the concept IFMG to examine the order of vertices in a social network group according to communication gaps., Competing Interests: Conflict of interestThe author declare that they have no conflict of interest., (© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021.)

Connectivity and strength has a major role in the field of network connecting with real world life. Complexity function is one of these parameter which has manifold number of applications in molecular chemistry and the theory of network. Firstly, this paper introduces the thought of complexity function of fuzzy graph with its properties. Second, based on the highest and lowest load on a network system, the boundaries of complexity function of different types of fuzzy graphs are established. Third, the behavior of complexity function in fuzzy cycle, fuzzy tree and complete fuzzy graph are discussed with their properties. Fourth, applications of these thoughts are bestowed to identify the most effected COVID-19 cycles between some communicated countries using the concept of complexity function of fuzzy graph. Also the selection of the busiest network stations and connected internet paths can be done using the same concept in a graphical wireless network system., Competing Interests: Conflict of interestThe authors declare that they have no conflict of interest., (© The Author(s), under exclusive licence to Springer Nature B.V. 2021.)

As an extension of intuitionistic fuzzy sets, the theory of picture fuzzy sets not only deals with the degrees of rejection and acceptance but also considers the degree of refusal during a decision-making process; therefore, by incorporating this competency of picture fuzzy sets, the goal of this study is to propose a novel hybrid model called picture fuzzy soft expert sets by combining picture fuzzy sets with soft expert sets for dealing with uncertainties in different real-world group decision-making problems. The proposed hybrid model is a more generalized form of intuitionistic fuzzy soft expert sets. Some novel desirable properties of the proposed model, namely, subset, equality, complement, union and intersection, are investigated together with their corresponding examples. Two well-known operations AND and OR are also studied for the developed model. Further, a decision-making method supporting by an algorithmic format under the proposed approach is presented. Moreover, an illustrative application is provided for its better demonstration, which is subjected to the selection of a suitable company of virtual reality devices. Finally, a comparison of the initiated method is explored with some existing models, including intuitionistic fuzzy soft expert sets.

In chemical graph theory, forgotten topological index or F -index plays a crucial role to collect information about the properties of chemical compounds. The k th generalized transformation graphs of a molecular graph preserve the entire information on the molecular topology contained in the relevant molecular graph. In this paper, some exact expressions of the F-index and its co-index for the k th generalized transformation graphs are obtained. Also, the Zagreb polynomials, Zagreb co-polynomials and their complements are computed., Competing Interests: The authors declare no conflict of interest., (© 2020 The Author(s).)

The theory of graphs are very useful tool in solving the combinatorial problems in different areas of computer science and computational intelligence systems. In this paper, we present a frame work to handle m -polar fuzzy information by combining the theory of m -polar fuzzy sets with graphs. We introduce the notion of weak self complement m -polar fuzzy graphs and establish a necessary condition for m -polar fuzzy graph to be weak self complement. Some properties of self complement and weak self complement m -polar fuzzy graphs are discussed. The order, size, busy vertices and free vertices of an m -polar fuzzy graphs are also defined and proved that isomorphic m -polar fuzzy graphs have same order, size and degree. Also, we have presented some results of busy vertices in isomorphic and weak isomorphic m -polar fuzzy graphs. Finally, a relative study of complement and operations on m -polar fuzzy graphs have been made. Applications of m -polar fuzzy graph are also given at the end.