Q-fractional fuzzy influence pair domination number to locate and control smog area

An intuitionistic fuzzy graph $\left(IFG\right)$ and its extensions could not handle the situation of the form ${\eta }_2=\{({ħ}_1,0.8,0.7),({ħ}_2,0.9,0.8),({ħ}_3,1,1)\}$ because $0.8+0.7=1.5>1$, $0.9+0.8=1.7>1$, and $1+1=2>1$. In this article, we proposed the concept of a q-fractional fuzzy influence graph $\left(q{f}_{r}FIG\right)$. A $q{f}_{r}FIG$ can indicate degrees of membership and non-membership 100% independently using the q-intercept of a straight line. We explore some ideas like a strongest q-fractional fuzzy influence pair $\left(SGq{f}_{r}FIP\right)$, strong q-fractional fuzzy influence pair $\left(Sq{f}_{r}FIP\right)$, weak q-fractional fuzzy influence pair $\left(Wq{f}_{r}FIP\right)$, q-fractional fuzzy influence cut-node $\left(q{f}_{r}FICN\right)$, q-fractional fuzzy influence bridge $\left(q{f}_{r}FIB\right)$, q-fractional fuzzy influence cut-pair $\left(q{f}_{r}FICP\right)$, and minimum $Sq{f}_{r}FIP$ domination number. These concepts, propositions, and theorems are explained using examples to strengthen our proposed graphical model. Finally, we present an application of a minimum $Sq{f}_{r}FIP$ domination number to locate and control the smog area in a $q{f}_{r}FIG$. Additionally, we present a comparative analysis of our proposed graphical model with an $IFG$, some multi-criteria decision-making techniques (EDAS, VIKOR, and TOPSIS methods), and some entropies to prove the validity and effectiveness of our proposed model. This comprehensive evaluation shows the applicability and effectiveness of our proposed approach.