Randić energies in decision making for human trafficking by interval-valued T-spherical fuzzy Hamacher graphs

The interval-valued ( $IV$) T-spherical fuzzy set $(IVTSFS)$ appears to be more effective and practical in dealing with uncertainty and ambiguity while dealing with various decision-making ( $DM$) problems than other fuzzy sets, for example, the q-rung ortho-pair fuzzy set, T-spherical fuzzy set, and picture fuzzy set. In real-life problems, where intuitionistic fuzzy sets, interval-valued intuitionistic fuzzy sets, or interval-valued picture fuzzy sets give unsatisfactory results, the $IVTSFS$ is a mathematical model that is used to deal with such problems where these method fails, and one can handle it efficiently by using $IVTSFS.$ If there is a different framework that has a base of four opinions — yes, no, abstains, and refusal — the $IVTSFS$ looks to be the most beneficial, and it is also proven to be so. In this article, an idea of interval-valued T-spherical fuzzy Hamacher graphs $(IVTSFHGs)$ is proposed, which is based on the Hamacher t-norm ( $TN$) and Hamacher t-conorm ( $TCN$). It provides enhanced discrimination and flexibility in uncertain environments by capturing a broader spectrum of hesitancy and indeterminacy, which other fuzzy models like interval-valued intuitionistic fuzzy set ( $IVIFS$) or picture fuzzy set ( $PFS$) may overlook. This article aims to examine the energy associated with the splitting of $IVTSFHGs,$ as well as the energy related to shadow $IVTSFHGs.$ Moreover, the Randić energy of $IVTSFHGs$ was presented, and an in-depth analysis of its essential outcomes was conducted. Additionally, this study introduces interval-valued T-spherical fuzzy Hamacher digraphs $(IVTSFHDGs)$ and explores their diverse outcomes. An algorithm involving $IVTSFHDGs$ was also studied, which is used to propose energies of $IVTSFHGs$ in $DM$, problems, and Hamacher aggregation operators. To validate the proposed results, a comparative study is conducted.