Evaluation of Home Loan Candidates Using Energy and Laplacian Energy in Hesitancy Fuzzy Zagreb Matrices

A hesitancy fuzzy graph is a type of generalized fuzzy graphs that reflects reluctance or uncertainty in interactions by assigning a variety of potential membership values rather than a single value to each vertex and edge. It offers an adaptable framework for simulating intricate systems with ambiguous, reluctant, or imperfect input. Topological measurements that capture and characterize a graph's structural features are called Zagreb indices, and they are obtained from vertex degrees. Although Zagreb indices and graph energies have been studied for intuitionistic fuzzy graphs, fuzzy graphs, and classical graphs, no work has been done to apply these ideas to hesitancy fuzzy graphs. The framework of hesitancy fuzzy Zagreb matrices is extended in this study by investigating their energy and demonstrating the concept of Laplacian energy. We propose upper and lower bounds for the energy of the hesitancy fuzzy Zagreb matrix and its Laplacian energy in the hypothetical context of choosing a house loan applicant, which is frequently impacted by variables like the CIBIL score. Additionally, the energy and Laplacian energy of the hesitancy fuzzy Zagreb matrix are compared.