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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.

$(\alpha,\beta)$ -Pythagorean fuzzy set is a very efficient way of dealing with uncertainty. In this article, we have introduced the notions of $(\alpha,\beta)$ - Pythagorean fuzzy subring and $(\alpha,\beta)$ - Pythagorean fuzzy ideal of a ring. Further, we have briefly described various results related to it. Also, we have discussed the level subring of an $(\alpha,\beta)$ - Pythagorean fuzzy subring. Moreover, we have studied the direct product and ring homomorphism of $(\alpha,\beta)$ - Pythagorean fuzzy subrings.

Pythagorean fuzzy environment is the modern tool for handling uncertainty in many decisions making problems. In this paper, we represent the notion of Pythagorean fuzzy subgroup (PFSG) as a generalization of intuitionistic fuzzy subgroup. We investigate various properties of our proposed fuzzy subgroup. Also, we introduce Pythagorean fuzzy coset and Pythagorean fuzzy normal subgroup (PFNSG) with their properties. Further, we define the notion of Pythagorean fuzzy level subgroup and establish related properties of it. Finally, we discuss the effect of group homomorphism on Pythagorean fuzzy subgroup.

An (α,β)-Pythagorean fuzzy environment is an efficient tool for handling vagueness. In this paper, the notion of relative subgroup of a group is introduced. Using this concept, the (α,β)-Pythagorean fuzzy order of elements of groups in (α,β)-Pythagorean fuzzy subgroups is defined and examined various algebraic properties of it. A relation between (α,β)-Pythagorean fuzzy order of an element of a group in (α,β)-Pythagorean fuzzy subgroups and order of the group is established. The extension principle for (α,β)-Pythagorean fuzzy sets is introduced. The concept of (α,β)-Pythagorean fuzzy normalizer and (α,β)-Pythagorean fuzzy centralizer of (α,β)-Pythagorean fuzzy subgroups are developed. Further, (α,β)-Pythagorean fuzzy quotient group of an (α,β)-Pythagorean fuzzy subgroup is defined. Finally, an (α,β)-Pythagorean fuzzy version of Lagrange's theorem is proved.

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.

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.

In mathematical chemistry, the topological indices with highly correlation factor play a leading role specifically for developing crucial information in QSPR/QSAR analysis. Recently, there exists a new graph invariant, namely, Y-index of graph proposed by Alameri as the sum of the fourth power of each and every vertex degree of that graph. The approximate range of the descriptors is determined by obtaining the bounds for the topological indices of graphs. In this paper, firstly, some upper bounds for the Y-index on trees with several types of domination number are studied. Secondly, some new bounds are also presented for this index of graphs in terms of relevant parameters with other topological indices. Additionally, a new idea on bounds for the Y-index by applying binary graph operations is computed.

In this study, from a tree with a quasi-spanning face, the algorithm will route Hamiltonian cycles. Goodey pioneered the idea of holding facing 4 to 6 sides of a graph concurrently. Similarly, in the three connected cubic planar graphs with two-colored faces, the vertex is incident to one blue and two red faces. As a result, all red-colored faces must gain 4 to 6 sides, while all obscure-colored faces must consume 3 to 5 sides. The proposed routing approach reduces the constriction of all vertex colors and the suitable quasi-spanning tree of faces. The presented algorithm demonstrates that the spanning tree parity will determine the arbitrary face based on an even degree. As a result, when the Lemmas 1 and 2 theorems are compared, the greedy routing method of Hamiltonian cycle faces generates valuable output from a quasi-spanning tree. In graph idea, a dominating set for a graph S=V,E is a subset D of V. The range of vertices in the smallest dominating set for S is the domination number (S). Vizing’s conjecture from 1968 proves that the Cartesian fabricated from graphs domination variety is at least as big as their domination numbers production. Proceeding this work, the Vizing’s conjecture states that for each pair of graphs S,L.

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 RFY=∑uv∈EYdYu+dYv−23, where dYu 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.

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.

The study of the inverse problem (IP) based on the topological indices (TIs) deals with the numerical relations to TIs. Mathematically, the IP can be expressed as follows: given a graph parameter/TI that assigns a non-negative integer value g to every graph within a given family G of graphs, find some G∈G for which TIG=g. It was initiated by the Zefirov group in Moscow and later Gutman et al. proposed it. In this paper, we have established the IP only for the Y-index, Gourava indices, second hyper-Zagreb index, reformulated first Zagreb index, and reformulated F-index since they are closely related to each other. We have also studied the same which is true for the molecular, tree, unicyclic, and bicyclic graphs.

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.

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.

In chemical graph theory, forgotten topological index or F-index plays a crucial role to collect information about the properties of chemical compounds. The kth 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 kth generalized transformation graphs are obtained. Also, the Zagreb polynomials, Zagreb co-polynomials and their complements are computed.

Picture fuzzy sets (PFSs) can be used to handle real-life problems with uncertainty and vagueness more effectively than intuitionistic fuzzy sets (IFSs). In the process of information aggregation, many aggregation operators under PFSs are used by different authors in different fields. In this article, a multi-attribute decision-making (MADM) problem is introduced utilizing harmonic mean aggregation operators with trapezoidal fuzzy number (TrFN) under picture fuzzy information. Three harmonic mean operators are developed namely trapezoidal picture fuzzy weighted harmonic mean (TrPFWHM) operator, trapezoidal picture fuzzy order weighted harmonic mean (TrPFOWHM) operator and trapezoidal picture fuzzy hybrid harmonic mean (TrPFHHM) operator. The related properties about these operators are also studied. At last, an MADM problem is considered to interrelate among these operators. Furthermore, a numerical instance is considered to explain the productivity of the proposed operators.

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.