Smarandache developed the idea of hypersoft set (HSS) theory as an extension of soft set (SS) theory. HSS provides a general mathematical framework for handling data that can be formulated as several trait-valued disjoint sets which blend to various traits. The major goal of this article is to lay the footing for supplying a new model called bipolar fuzzy hypersoft sets (BFHSSs) by linking both fuzzy sets (FSs) and HSSs under bipolarity property. By using positive and negative membership functions and multi-argument functions, these structures work best for testing uncertainty. This makes them better at solving real-world problems, especially ones that have both good and bad sides. This paper also has different operations for BFHSSs, such as absolute BFHSS, null BFHSS, complement, subset, union, intersection, and their related properties. Moreover, operations like OR and AND for BFHSS have been instituted. Some properties are demonstrated, and some numerical examples are given to illustrate the mechanism of using these tools. Finally, these tools are applied in the decision-making process based on an algorithm that is built.

In this paper, we discuss the energy of the commuting graph. The vertex set of the graph is dihedral groups and the edges between two distinct vertices represent the commutativity of the group elements. The spectrum of the graph is associated with the Seidel Laplacian and Seidel signless Laplacian matrices. The results are similar to the well-known fact that the energies are not odd integers. We also highlight the relation that the Seidel signless Laplacian energy is never less than Seidel Laplacian energy. Ultimately, we classify the graphs according to the energy value as the hyperenergetic.

American scientific publishing group

A fundamental feature of spectral graph theory is the correspondence between matrix and graph. As a result of this relation, the characteristic polynomial of the graph can be formulated. This research focuses on the power graph of dihedral groups using degree-based matrices. Throughout this paper, we formulate the characteristic polynomial of the power graph of dihedral groups based on seven types of graph matrices which include the maximum degree, the minimum degree, the greatest common divisor degree, the first Zagreb, the second Zagreb, the misbalance degree, and the Nirmala matrices.

The prime coprime graph is a graph in which every pair of vertices is adjacent if and only if the greatest common divisor of the order of both vertices is equal to one or relatively prime. This study aims to analyze the general formula of the first and second Zagreb, and hyper Zagreb indices of prime coprime graphs for a group of integers modulo power of primes. The correlation between these indices is found to have a strong significant correlation; in other words, if one index increases, then the other indices also increase with a coefficient correlation greater than 0.95. We also demonstrate this correlation in one equation and our results improve the understanding of Zagreb-based indices.

American scientific publishing group