@article{Ahmad2025, 
author = {Ali Ahmad and Humera Rashid and Hamdan Alshehri and Muhammad Kamran Jamil and Haitham Assiri},
title = {Randić energies in decision making for human trafficking by interval-valued T-spherical fuzzy Hamacher graphs},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {4},
pages = {9697-9747},
keywords = {Randić energies, human trafficking, interval-valued T-spherical fuzzy Hamacher graphs, fuzzy graph},
url = {https://www.sciopen.com/article/10.3934/math.2025446},
doi = {10.3934/math.2025446},
abstract = {The interval-valued (   I  V) T-spherical fuzzy set    (  I  V  T  S  F  S  ) appears to be more effective and practical in dealing with uncertainty and ambiguity while dealing with various decision-making (   D  M) 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    I  V  T  S  F  S is a mathematical model that is used to deal with such problems where these method fails, and one can handle it efficiently by using    I  V  T  S  F  S  . If there is a different framework that has a base of four opinions — yes, no, abstains, and refusal — the    I  V  T  S  F  S 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    (  I  V  T  S  F  H  G  s  ) is proposed, which is based on the Hamacher t-norm (   T  N) and Hamacher t-conorm (   T  C  N). 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 (   I  V  I  F  S) or picture fuzzy set (   P  F  S) may overlook. This article aims to examine the energy associated with the splitting of    I  V  T  S  F  H  G  s  , as well as the energy related to shadow    I  V  T  S  F  H  G  s  . Moreover, the Randić energy of    I  V  T  S  F  H  G  s was presented, and an in-depth analysis of its essential outcomes was conducted. Additionally, this study introduces interval-valued T-spherical fuzzy Hamacher digraphs    (  I  V  T  S  F  H  D  G  s  ) and explores their diverse outcomes. An algorithm involving    I  V  T  S  F  H  D  G  s was also studied, which is used to propose energies of    I  V  T  S  F  H  G  s in    D  M, problems, and Hamacher aggregation operators. To validate the proposed results, a comparative study is conducted.}
}