Optimizing graphical network design using fuzzy hypersoft topologies and multi-criteria VIKOR decision algorithms

The accumulated data is displayed in the form of a figure called a graph. The notion of fuzziness was developed in graph theory to address issues not resolved by crisp graph theory. However, fuzzy soft ideas in graph theory, which is a parameterized family and provides more exact and generalized solutions to answer vague and ambiguous problems, can yield more generalized conclusions. Fuzzy hypersoft is another broader and more generic soft set, which is the Cartesian product of disjoint attribute-valued sets corresponding to distinct attributes, in which mapping is transformed into a multi-attribute. The number of fuzzy hypersoft subgraphs is equal to the cardinality of the set obtained after taking the Cartesian product of disjoint attributed valued sets, which could be applied in graph theory to get surprising results. The novelty of this manuscript is defining topological numbers for the first time in a fuzzy hypersoft environment. We have also initiated the definitions of path graph, cycle graph, complete graph, wheel graph, and star graph, and calculated the degrees of nodes of each graph in a fuzzy hypersoft environment. Then, we have derived some generalized results for these fuzzy hypersoft families of graphs for the first Zagreb number, the second Zagreb number, and the Randic number. This is also a novel approach that we have converted the decision-making algorithm, vizierkriterijumsko kompromisno rangiranje (VIKOR), into a fuzzy hypersoft framework. Our objectives are to lessen the complexity in the approaches and to establish a solid relationship with the multi-criteria decision-making procedures. It is intriguing that the decision-making issue can be addressed with hypersoft theory without the restrictions on the decision-maker’s choice of values. In the end, we have presented an application to highlight the generic nature of the fuzzy hypersoft environment. We also have examined the best graphical network to maximize profit for a four-partner corporation, taking into account fuzzy hypersoft topological numbers as decision-makers. Then, VIKOR in the fuzzy hypersoft framework is used for the same purpose. The acquired results are enough to prove that the graphical network for yielding maximum profit is the same by applying MCDM as well as by applying the formulas of different topological numbers. Furthermore, we have presented the AHP method’s methodology and eventually employed it to determine the top-ranked graphical network among commercial enterprises. Furthermore, a comparison has been made between fuzzy soft topological numbers and fuzzy hypersoft topological numbers.