On Graph Structures in Fuzzy Environment Using Optimization Parameter

This paper comprises the introduction of weighted mean products of fuzzy graph structures (
FGSs
) to construct weighted mean fuzzy graph structures (
WMFGSs
) with the help of optimization parameter, and establish some novel results after validating with examples, accordingly. The notions of regular and
$m\mu _{k}$
-regular
FGSs
are described, where
$m\in$
]0, 1] represents the degree of all vertices in
$\widetilde {G}$
under mapping
$\mu _{k}$
, and develop certain properties of regular
WMFGSs
. In addition, we create a flowchart to present common application procedures of fuzzy graphical frameworks to classify the advanced city out of some important Pakistani cities subject to certain parameters.