Multi attribute decision-making and interval-valued picture ( S , T )-fuzzy graphs

In this manuscript, we introduce the notion of interval-valued picture ( S , T )-fuzzy graphs (IVP-( S , T )-fuzzy graphs) which is the interplay between ( S , T )-norms and picture fuzzy graphs. Interval-valued picture ( S , T )-fuzzy graphs (IVP-( S , T )-fuzzy graphs) is also the extension of the picture fuzzy graphs. Since interval-valued picture fuzzy sets (IVPFSs) is the most extended form of the fuzzy sets to deal uncertainties, interval-valued picture ( S , T )-fuzzy graphs (IVP-( S , T )-fuzzy graphs) would be more efficient to deal with the problems containing vagueness. For the sake of investigation, firstly we introduce and apply various operations like union, join, cartesian product, direct product, lexicographic product, ring sum, complement etc to interval-valued picture ( S , T )-fuzzy graphs. Then, we study the structural properties of interval-valued picture ( S , T )-fuzzy graphs (IVP-( S , T )-fuzzy graphs) through homomorphism, co-weak homomorphism, isomorphism etc. Afterwards, we initiate different types of interval-valued picture ( S , T )-fuzzy graphs such as regular, total regular, constant etc. Moreover, we also explore some relationships among different types of interval-valued picture ( S , T )-fuzzy graphs. Finally, we provide an application of interval-valued picture ( S , T )-fuzzy graphs (IVP-( S , T )-fuzzy graphs) towards multiple attribute decision-making (MADM).