The fuzzy linguistic approach provides favorable outputs in several areas, whose description is relatively qualitative. The encouragement for the utilization of sentences or words instead of numbers is that linguistic characterizations or classifications are usually less absolute than algebraic or arithmetical ones. In this research article, we animate the m-polar fuzzy (mF) linguistic approach and elaborate it with real life examples and tabular representation to develop the affluence of linguistic variables based on mF approach. As an extension of the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method, we develop an m-polar fuzzy linguistic TOPSIS approach for multi-criteria group decision-making (MCGDM). It is used to evaluate the best alternative, to get more authentic and comparable results and to handle the real life problems of having multi-polar information in terms of linguistic variables and values. In this approach decision-makers contribute their estimations in the form of linguistic term sets. To show the efficiency and compatibility of the proposed approach, we compare it with the m-polar fuzzy linguistic ELECTRE-I (Elimination and Choice Translating Reality) approach. Finally, we draw a flow chart of our proposed approach as an algorithm and generate a computer programming code.

A two-mode network is a type of network in which nodes can be divided into two sets in such a way that links can be established between different types of nodes. The relationship between two separate sets of entities can be modeled as a bipartite network. In computer networks data is transmitted in form of packets between source to destination. Such packet-switched networks rely on routing protocols to select the best path. Configurations of these protocols depends on the network acquirements; that is why one routing protocol might be efficient for one network and may be inefficient for a other. Because some protocols deal with hop-count (number of nodes in the path) while others deal with distance vector. This paper investigates the minimum transmission in two-mode networks. Based on some parameters, we obtained the minimum transmission between the class of all connected n-nodes in bipartite networks. These parameters are helpful to modify or change the path of a given network. Furthermore, by using least squares fit, we discussed some numerical results of the regression model of the boiling point in benzenoid hydrocarbons. The results show that the correlation of the boiling point in benzenoid hydrocarbons of the first Zagreb eccentricity index gives better result as compare to the correlation of second Zagreb eccentricity index. In case of a connected network, the first Zagreb eccentricity index ${\xi }_1(\aleph )$ is defined as the sum of the square of eccentricities of the nodes, and the second Zagreb eccentricity index ${\xi }_2(\aleph )$ is defined as the sum of the product of eccentricities of the adjacent nodes. This article deals with the minimum transmission with respect to ${\xi }_i(\aleph )$, for $i=1,2$ among all n-node extremal bipartite networks with given matching number, diameter, node connectivity and link connectivity.

Coronoid systems actually arrangements of hexagons into six sides of benzenoids. By nature, it is an organic chemical structure. Hollow coronoids are primitive and catacondensed coronoids. It is also known as polycyclic conjugated hydrocarbons. The mathematical study of chemicals is of great interest to different specialties researchers. While graph theory always played a significant role to make chemical structures understandable and blessed with applications also. After transforming the chemical structure into a graph, one can implement different theoretical and implicative studies on structures. Metric dimension is considered as one of the most studied and implicative parameters of graph theory. In this concept, few suggested vertices are chosen such as the remaining vertices have unique locations or identifications. In this study, we discussed different metric-based parameters for the hollow coronoid structure.

Parameter reduction is a very important technique in many fields, including pattern recognition. Many reduction techniques have been reported for fuzzy soft sets to solve decision-making problems. However, there is almost no attention to the parameter reduction of bipolar fuzzy soft sets, which take advantage of the fact that membership and non-membership degrees play a symmetric role. This methodology is of great importance in many decision-making situations. In this paper, we provide a novel theoretical approach to solve decision-making problems based on bipolar fuzzy soft sets and study four types of parameter reductions of such sets. Parameter reduction algorithms are developed and illustrated through examples. The experimental results prove that our proposed parameter reduction techniques delete the irrelevant parameters while keeping definite decision-making choices unchanged. Moreover, the reduction algorithms are compared regarding the degree of ease of computing reduction, applicability, exact degree of reduction, applied situation, and multi-use of parameter reduction. Finally, a real application is developed to describe the validity of our proposed reduction algorithms.

Geometric arrangements of hexagons into six sides of benzenoids are known as coronoid systems. They are organic chemical structures by definition. Hollow coronoids are divided into two types: primitive and catacondensed coronoids. Polycyclic conjugated hydrocarbon is another name for them. Chemical mathematics piques the curiosity of scientists from a variety of disciplines. Graph theory has always played an important role in making chemical structures intelligible and useful. After converting a chemical structure into a graph, many theoretical and investigative studies on structures can be carried out. Among the different parameters of graph theory, the dimension of edge metric is the most recent, unique, and important parameter. Few proposed vertices are picked in this notion, such as all graph edges have unique locations or identifications. Different (edge) metric-based concept for the structure of hollow coronoid were discussed in this study.

In a few years, hesitant fuzzy sets (HFSs) have had an impact on several different areas of decision science. However, a number of researches have utilized the Elimination and choice translating reality (ELECTRE) methods to determine the multi-criteria decision-making (MCDM) problems with hesitant information. The aim of this research article is to develop new multi-criteria group decision-making (MCGDM) methods, such as the m-polar hesitant fuzzy ELECTRE-I (mHF ELECTRE-I) method and hesitant m-polar fuzzy ELECTRE-I (HmF ELECTRE-I) method. Proposed MCGDM techniques based on the hybrid models, m-polar hesitant fuzzy sets (mHFS-sets) and hesitant m-polar fuzzy sets (HmF-sets), which are the natural generalizations of HFSs and m-polar fuzzy sets (mF sets). These models enable us to deal with multipolar information under hesitancy. We use the proposed methods to deal the complex problems in which the membership degree of an element of given set uses the m different numeric and fuzzy values, to rank all the alternatives and to determine the best alternative. We present two practical examples that illustrate the procedure of the proposed methods. We also discuss the differences and comparative analysis of the proposed methods. Finally, we develop an algorithm that implements our decision-making procedures by using computer programming.