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The Pythagorean fuzzy environment is a modern way of depicting uncertainty. The concept of Pythagorean fuzzy semi-level subgroups of any group is described in this paper. The Pythagorean fuzzy order of an element in a Pythagorean fuzzy subgroup is introduced and established various algebraic attributes. The relation between the Pythagorean fuzzy order of an element of a group and the order of that group is established. The Pythagorean fuzzy normalizer and Pythagorean fuzzy centralizer of Pythagorean fuzzy subgroups are discussed. Further, the concept of Pythagorean fuzzy quotient group and the index of a Pythagorean fuzzy subgroup are defined. Finally, a framework is developed for proving Lagrange's theorem in Pythagorean fuzzy subgroups. [ABSTRACT FROM AUTHOR]

Cubic fuzzy graphs (CFGs) offer greater utility as compared to interval-valued fuzzy graphs and fuzzy graphs due to their ability to represent the degree of membership for vertices and edges using both interval and fuzzy number forms. The significance of these concepts motivates us to analyze and interpret intricate networks, enabling more effective decision making and optimization in various domains, including transportation, social networks, trade networks, and communication systems. This paper introduces the concepts of vertex and edge connectivity in CFGs, along with discussions on partial cubic fuzzy cut nodes and partial cubic fuzzy edge cuts, and presents several related results with the help of some examples to enhance understanding. In addition, this paper introduces the idea of partial cubic α-strong and partial cubic δ-weak edges. An example is discussed to explain the motivation behind partial cubic α-strong edges. Moreover, it delves into the introduction of generalized vertex and edge connectivity in CFGs, along with generalized partial cubic fuzzy cut nodes and generalized partial cubic fuzzy edge cuts. Relevant results pertaining to these concepts are also discussed. As an application, the concept of generalized partial cubic fuzzy edge cuts is applied to identify regions that are most affected by trade deficits resulting from street crimes. Finally, the research findings are compared with the existing method to demonstrate their suitability and creativity. [ABSTRACT FROM AUTHOR]

The first entire Zagreb index (FEZI) is a graph parameter that has proven to be essential in various real-life scenarios, such as networking businesses and traffic management on roads. In this research paper, the FEZI was explored for a variety of fuzzy graphs, including star, firefly graph, cycle, path, fuzzy subgraph, vertex elimination, and edge elimination. This study presented several results, including determining the relationship between two isomorphic fuzzy graphs and between a path and cycle (connecting both end vertices of the path). This research also deals with the analysis of α -cut fuzzy graphs and establishes bounds for some fuzzy graphs. To apply these findings to modern life problems, the research team utilized the results to identify areas that require more development in internet systems. These results have practical implications for enhancing the efficiency and effectiveness of internet systems. The conclusion drawn from this research can be used to inform future research and aid in the development of more efficient and effective systems in various fields. [ABSTRACT FROM AUTHOR]

The inverse minimum spanning tree (IMST) problem is an inverse optimization problem in which one makes the least modification to the edge weights of a predetermined spanning tree, to make it the minimum spanning tree with respect to new edge weights. For a deterministic environment, the problem has been extensively studied. In an uncertain environment, the problem has been studied previously using stochastic edge weights or fuzzy edge weights. However, in the absence of enough data, approximation of a random variable is not possible. Further, the unobservable nature of edge weights means that assignment of fuzzy weights is also not possible. In this situation, the assignment of edge weights is done based on belief degree of some experts in the field. To deal with the problem of belief degree, the uncertainty theory is mostly suited. In this paper, two specific models for inverse minimum spanning tree are initiated, taking rough variables and uncertain normal variables as edge weights. Based on the properties of uncertainty, two specific models are formulated for the inverse minimum spanning tree problem. The models are converted to their equivalent deterministic models, which are solved by some standard optimization method. A numerical example is given to illustrate the model and its solution. [ABSTRACT FROM AUTHOR]

The valency based topological indices (TIs) are defined by algebraic functions as the chemical graph theory tools in which the structural parameters are used as input and the output is related with the topology of chemical species. In theoretical chemistry, the TIs are mainly used to investigate/develop the QSAR and QSPR investigations of the molecular graphs. The Reformulated F-index or RFI is one such kind of TIs. The RFI for a molecular graph Y is defined as R F Y = ∑ u v ∈ E Y d Y u + d Y v − 2 3 , where d Y u is the degree of a vertex u in Y. In this paper, we study the RFI for vertex and edge version of F-join of given general graphs which are related to subdivision and total graph. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Topological indices (TIs) are expressed by constant real numbers that reveal the structure of the graphs in QSAR/QSPR investigation. The reformulated second Zagreb index (RSZI) is such a novel TI having good correlations with various physical attributes, chemical reactivities, or biological activities/properties. The RSZI is defined as the sum of products of edge degrees of the adjacent edges, where the edge degree of an edge is taken to be the sum of vertex degrees of two end vertices of that edge with minus 2. In this study, the behaviour of RSZI under graph operations containing Cartesian product, join, composition, and corona product of two graphs has been established. We have also applied these results to compute RSZI for some important classes of molecular graphs and nanostructures. [ABSTRACT FROM AUTHOR]

Fuzzy set is a modern tool for depicting uncertainty. This paper introduces the concept of fuzzy sub e-group as an extension of fuzzy subgroup. The concepts of identity and inverse are generalized in fuzzy sub e-groups. Every fuzzy subgroup is proven to be a fuzzy sub e-group, but the converse is not true. Various properties of fuzzy sub e-groups are established. Moreover, the concepts of proper fuzzy sub e-group and super fuzzy sub e-group are discussed. Further, the concepts of fuzzy e-coset and normal fuzzy sub e-group are presented. Finally, we describe the effect of e-group homomorphism on normal fuzzy sub e-groups. [ABSTRACT FROM AUTHOR]

A Pythagorean fuzzy set (PFS) is a very efficient and powerful tool for handling uncertainty and vagueness. In this paper, we present the notion of (α,β) -Pythagorean fuzzy set (PFS) as a generalisation of Pythagorean fuzzy set (PFS). We propose a new fuzzy subgroup, called (α,β) -Pythagorean fuzzy subgroup (PFSG). We investigate properties of our proposed fuzzy subgroup. We present the concept of (α,β) -Pythagorean fuzzy coset (PFC) and (α,β) -Pythagorean fuzzy normal subgroup (PFNSG). Furthermore, we define (α,β) -Pythagorean fuzzy level subgroup (PFLSG) and establish properties of it. [ABSTRACT FROM AUTHOR]

Recently, m-polar fuzzy graph (mPFG) becomes a growing research topic as it is the generalization of fuzzy graph. In this paper, at first α-strong mPF arc, β-strong mPF arc, δ-strong mPF arc and δ ∗-strong mPF arc in an mPFG are defined. An application of decision making using strong path is also given at the end. [ABSTRACT FROM AUTHOR]

The third leap Zagreb index of a graph G is denoted as L M 3 (G) and is defined as L M 3 (G) = ∑ b ∈ V (G) d (b / G) d 2 (b / G) , where d 2 (b / G) and d (b / G) are the 2-distance degree and the degree of the vertex b in G , respectively. The first, second and third leap Zagreb indices were introduced by Naji et al. [A. M. Naji, N. D. Soner and I. Gutman, On leap Zagreb indices of graphs, Commun. Combin. Optim. 2(2) (2017) 99–117] in 2017. In this paper, the behavior of the third leap Zagreb index under several graph operations like the Cartesian product, Corona product, neighborhood Corona product, lexicographic product, strong product, tensor product, symmetric difference and disjunction of two graphs is studied. [ABSTRACT FROM AUTHOR]

This paper deals with fuzzy permutation graph which is introduced as a fuzzified model of a crisp permutation graph. We have defined two types of complements of a fuzzy permutation graph viz. p-complement and f-complement. We have considered various examples to encounter the existence and non-existence of the f-complement graph as a fuzzy permutation graph and established a sufficient condition for the existence of the same. Also, we have calculated the possible number of fuzzy permutation graphs that can represent the f-complement of a fuzzy permutation graph for a given permutation depending on the edge-membership values of the fuzzy permutation graph. We have also introduced and characterized fuzzy comparability graph. Lastly, we have discussed a real-life application of fuzzy permutation graph. [ABSTRACT FROM AUTHOR]

In this paper, the embedding of intuitionistic fuzzy graphs which are constructed on the surface of spheres is introduced. The intuitionistic fuzzy genus graph with its genus value, strong and weak intuitionistic fuzzy genus graph are defined. The isomorphism properties on intuitionistic fuzzy genus graph are discussed. The relation between planarity value and genus value of an intuitionistic fuzzy graph is established. Also, Euler polyhedral equation is established in terms of genus value of the intuitionistic fuzzy genus graph. Finally, an application of intuitionistic fuzzy genus graph is given in topological surface. [ABSTRACT FROM AUTHOR]

In this paper, the embedding of m-polar fuzzy graphs which constructed on the surface of spheres is introduced. The m-polar fuzzy genus graphs with its genus value, strong and weak m-polar fuzzy genus graph are defined. Also isomorphism properties on m-polar fuzzy genus graph are discussed. The relation between planarity value and genus value of m-polar fuzzy graph is established. Euler polyhedral equation is established in terms of genus value of the m-polar fuzzy genus graph. Finally, an application of m-polar fuzzy genus graph is given in topological surface. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]